國 立 交 通 大 學
應用數學系
碩
士
論
文
半模函式與 Shimura 對應
Half Integral Weight Modular Forms and
Shimura correspondence
研 究 生：林家銘
指導老師：楊一帆 教授
半模函式與 Shimura 對應
Half Integral Weight Modular Forms and
Shimura correspondence
研 究 生：林家銘 Student：
Jia-Ming Lin
指導教授：楊一帆 教授 Advisor：
Professor Yifan Yang
國 立 交 通 大 學
應 用 數 學 系
碩 士 論 文
A Thesis
Submitted to Department of Applied Mathematics
College of Science
National Chiao Tung University
in Partial Fulfillment of the Requirements
for the Degree of
Master
in
Applied Mathematics
June 2010
Hsinchu, Taiwan, Republic of China
i
半模函式與 Shimura 對應
學生：林家銘 指導老師：楊一帆 教授
國立交通大學應用數學系(研究所)碩士班
摘 要
1973 年，G.Shimura 利用 theta-函數來定義半模函式，並且寫出
Hecke 算子作用在半模函式上的一般項。他發現 Hecke 算子作用在半模函式
上的特徵值與整模函式的特徵值有對應關係，這就是所謂的 Shimura 對
應。
另一方面，eta-函數是一個 (1/2) weight 的模函式在 Shimura 的定
義之下。本篇論文當中主要探討用 eta-函數所定義出來的半模函式空間，
這也許會使我們能夠證明出一些分析函數的同於式。歷史上這種由 eta-函
數所定義出來的半模函式空間的研究開使於 Li Guo 和 Ken Ono 在“The
partition function and the arithmetic of certain modular
L-functions＂中，並且證明了在某些例子中這種子空間是同構於一個整
模函式的子空間，而這個整模函式的子空間是一些算子的不變子空間。我
們現在把他們的結果更一般化，並且算出對應空間的維度。不過由於時間
的關係，我們仍無法算出對於一般 Hecke 算子的 trace formula，也許在將
來的日子裡會有機會把他算出來。
ii
Half Integral Weight Modular Forms and
Shimura correspondence
Student：Jia Ming Lin Advisor：Professor Yifan Yang
Department ( Institute ) of Applied Mathematics
National Chiao Tung University
ABSTRACT
In 1973, G.Shimura defined modular forms of half-integral weight
by using theta-function. He showed that there are Hecke operators
on half-integral weight modular forms, and he found that there is a
correspondence between each eigenvalue for Hecke operator for
integral weight modular form and half-integral weight modular form.
And it is the so-called Shimura correspondence.
On the other hand, eta-function is a modular form of weight (1/2)
in Shimura′s sense. In this paper, we study the space of half-integral
weight modular forms defined by eta-function, so that we may find
some congruence of partition functions. Historically, these spaces
were first studied by Li Guo and Ken Ono in their paper“The
partition function and the arithmetic of certain modular
iii
isomorphic to space of integral-weight modular forms which is
eigenspace of some operators. Now we make more general results,
and we compute the dimensions in our cases. For the isomorphism,
we try to prove it by using trace formula, but it is so complicated that
we have not figured it out yet.
iv
誌 謝
首先，本篇論文的完成要感謝楊一帆老師給我這個機會接觸這方面
的知識，使我能夠學習到數論的元素與應用，若將來能有幸在此領域繼
續研讀，那麼這些知識一定是生涯中最重要的基礎。
最後給予所有朋友某種程度上的感謝。
林家銘
謹誌于交通大學
2010年7月
v
目錄
中文提要
………
i
英文提要
………
ii
致謝
………
iv
目錄
………
v
1.Introduction
………
1
2. Standard Definition and Background
………
3
2.1 Notations
………
32.2 Congruence Subgroup
0(
N
)
………
32.3 Atkin-Lehner involutions
………
52.4 Modular forms of integral weight
………
52.5 Hecke operators on integral weight modular forms
………
62.6 Modular forms of half integral weight
………
82.7 Hecke operator Tm on Gk/2(M,
)………
11 2.8 Shimura correspondence………
123.Explicit formulas
………
15
3.1 Dimension of St(
0(2),
1) and St(
0(3),
2)………
15 3.2 Dimension of St(
0(6),
1,
2)………
184.Results
………
25
4.1 Sr,s(
0(3))………
25 4.2 Sr,s(
0(2))………
35Bibliography
……… 39
Chapter 1
Introduction
In 1973, G.Shimura[4] laid the foundations of a theory of half integral weight modular forms with level M is always divisible by 4. We consider the space of cusp forms denoted by Sk/2(M, χ) mainly, where k is a positive odd number and
χ is a Dirichlet character. Firstly, Shimura showed that there are Hecke operations Tn2 for every natural number n with
gcd(n, M ) = 1. Secondly, in the Main Theorem and its corollary, Shimura associated half integral weight modular forms with modular forms of integral weight. It is the so-called Shimura correspondence. In S.Niwa’s paper [5], he proved that Sk/2(M, χ) is always isomorphic to Sk−1(M/2).
Later, W. Kohnen in his paper[7] looked for a subspace which corresponds to the space of cusp forms of weight 2k on Γ0(M/4), where M/4 is square free. Elements contained in the subspace are cusp forms and with Fourier expansions of
the form
X
n≥1,(−1)kn≡0,1(4)
a(n)qn,
and he denoted this subspace by Sk+1/2+ (M, χ). And we can introduce Hecke operators T+(n2) on Sk+1/2+ (M, χ) for all n prime to M . He set up a theory of newforms similar to Atkin-Lehner-Li-Miyake. There is a canonically defined subspace Sk+1/2new (M, χ) ⊂ Sk+1/2+ (M, χ) and a canonical decomposition
Sk+1/2+ (M, χ) = M
r,d≥1,rd|N
Sk+1/2new (d, χ)|U (r2)
(where U (r2) is the operator replacing the nth Fourier coefficient of a modular form by its r2nth one, and N = M/4), and
Snew
k+1/2(M, χ) isomorphic to space of newforms S new
2k (N ) ⊂ S2k(N ) (space of cusp forms with weight 2k over Γ0(N ))
as Hecke modules. In particular, we have a strong ”multiplicity 1 theorem” for Sk+1/2new (M, χ). See Remark 2.8.1. In this thesis, we focus on a special subspace of Sk+1/2(Γ0(576)), defined by
Sr,s(Γ0(N )) := {η(24τ )rf (24τ ) : f (τ ) ∈ Ms(Γ0(N ))},
where N is a positive integer. It is known that η(24τ ) is a weight 12modular form on Γ0(576) with character χ12. Y.Yang
proved in [8] that this subspace is an invariant subspace of Sk+1/2(Γ0(576)) under the action of the Hecke algebra when
N = 1. Then he discovered some new congruences of the the partition function p(n) by applying Hecke operators on the subspace Sr,s. A remarkable result of [8] is
p m` 2uK−1n + 1 24 ≡ 0 mod m, 1
where m ≥ 13 is a prime number, K is a positive integer determined by Hecke operators applying on the subspace Sr,s(Γ0(1)), n is positive integer depend on the Hecke operator, and u is any positive integer.
Our main result is concerned with the space Sr,s(Γ0(3)). We compute some examples in Maple, and conjecture that
Sr,s(Γ0(3) and Sr,s(Γ0(2) are also an invariant subspace of Sk/2(Γ0(576)) for r = 1, 5 and 7, s = 2, 4, 6, 8, and some
Hecke operators. Moreover, Yang conjecture that Sr,s(Γ0(1) isomorphic to a space of newforms of integral weight(see
Proposition 3.2.1) as Hecke module when r = 1, 5, 7, 11, 13, 17, 19, 23. And here we also make a similar conjecture, but in our case, we can only find the correspondences for r = 1, 5, 7. For r = 11, 13, 17, 19, 23 we will check our conjecture fails by computing dimensions.
The invariance of Sr,sin our cases(N = 2, 3) can most likely be proved by a way similar to the proof of theorem 2 in
[8], but we have not work it out. One of the key points in the proof is the choice of h(τ ) = η(`2τ )24−rg(τ /24) = η(`2τ )24−τη(τ )rf (τ ), where l is a prime number. When N = 3 we have to make some modification on h.
And now in this thesis, we observe some Hecke operator acting on the basis of Sr,sand claim invariance by checking
Fourier coefficients(see example 4.1.1).
Furthermore Sr,s(Γ0(2)) and Sr,s(Γ0(3)) isomorphic to a space of newforms of weight 2s+r −1 on some congruence
Chapter 2
Standard Definition and Background
2.1
Notations
Z : set of integers. H : upper half plane. N : positive integer. p : prime number.
SL2(Z) : special linear group over Z of dimension 2.
RΓ: fundamental domain of congruence subgroup Γ in SL2(Z).
2.2
Congruence subgroup Γ
0(N )
If N is any positive integer we define Γ0(N ) to be the setΓ0(N ) = a b c d ∈ SL2(Z) : N |c . It is a subgroup of SL2(Z) of finite index.
In particular, if p is a prime and let Sτ = −1/τ and T τ = τ + 1 be the generators of SL2(Z) , then for every V in SL2(Z),
but not in Γ0(p), there exists an element P ∈ Γ0(p) and an integer k, 0 ≤ k < p, such that
V = P STk. Let R be the fundamental domain of SL2(Z). Then
RΓ0(p)= R ∪ p−1 [ k=0 STk(R), where p is a prime.
Generally, we can also compute the index of Γ0(N ) in SL2(Z) and find the coset representations. Explicitly,
[SL2(Z) : Γ0(N )] = 2N
Y
p|N
(1 + 1/p). 3
and let γj=
aj bj
cj dj
∈ SL2(Z), j = 1, 2. The following three statement are equivalent:
1. The right cosets Γ0(N )γ1and Γ0(N )γ1are equal.
2. c1d2≡ c2d1 mod N .
3. There exist an integer r with gcd(r, N ) = 1 such that c1≡ rc2and d1≡ rd2 mod N .
Then we have
Theorem 2.2.1. Let S be the set of pairs (c, d) ∈ Z2withgcd(c, d, N ) = 1. Define an equivalence relation on S by
(c1, d1) ∼ (c2, d2) if and only if c1d2≡ c2d1 mod N . Then the coset representations of Γ0(N ) SL2(Z) is
∗ ∗ c d ∈ SL2(Z) : (c, d) ∈ S/ ∼ , where∗ ∗ c d
means the matrixa b c d
∈ SL2(Z).
Theorem 2.2.2. A set of inequivalent cusps for Γ0(N ) is given by
na
c : c|N, a = 1, . . . , gcd(c, N/c), gcd(a, c) = 1 o
. Hence the number of inequivalent cusps is
X
c|N
φ(gcd(c, N/c)), whereφ is the Euler totient function.
Theorem 2.2.3. (a) The number v2of inequivalent elliptic points of order 2 forΓ0(N ) is equal to the number of solutions
ofx2+ 1 = 0 in Z/N Z. That is, when 4|N , v2=0, and when4 - N ,
v2= Y p|N,p odd 1 + −1 p , where −1 p
is the Jacobi symbol.
(b) The number v3of inequivalent elliptic points of order 3 forΓ0(N ) is equal to the number of solutions of x2+x+1 = 0
in Z/N Z. That is, when 9|N , v3=0, and when9 - N ,
v3= Y p|N 1 + −3 p , where −3 p
2.3. ATKIN-LEHNER INVOLUTIONS 5
2.3
Atkin-Lehner involutions
Let N be a positive integer ≥ 2. Let n be a divisor of N such that gcd(n, N/n) = 1. The elements in ωn= 1 √ n an b cN dn , adn2− bcN = n
are the Atkin-Lehner involutions, which normalize Γ0(N ). The set of Γ0(N ) union all possible Atkin-Lehner involutions
is denoted by Γ∗0(N ).
2.4
Modular forms of integral weight
Here we let k be a positive integer. Let α =a b c d
be an element of GL+2(R) (general linear group over R with positive determinant).
Definition 2.4.1. Let f be a meromorphic function on H and α as above. Then we define f (τ )|[α]k= det(α)k/2(cτ + d)−kf (ατ ) (k ∈ N).
Then we call a holomorphic function on H is a modular form of weight k with respect to a congruence subgroup Γ of SL2(Z) if f satisfies:
1. f (τ )|[α]k= f (τ ), where α ∈ Γ and τ ∈ H,
2. f is holomorphic at every cusp of Γ. Since1 1
0 1
∈ Γ, f (τ + 1) = f (τ )|[α]k = f (τ ). Hence f has a Fourier expansion of the form ∞
X
n=0
anqn,
where q = e2πiτ and an ∈ H. If f(τ ) is vanish at all cusps, a0 = 0 in the Fourier expansion, and we call such f cusp
form.
We denote by
1. Mk(Γ) the set of all holomorphic modular forms.
2. Sk(Γ) the set of all cusp forms.
And there are dimension formulas:
Theorem 2.4.2. Let Γ be a subgroup of finite index of SL2(Z). Assume that the genus of X(Γ)(the compactified modular
curveΓ \ H∗) is g. Let c be the number of inequivalent cusps of Γ, and e
1, . . . , erbe the orders of inequivalent elliptic
points. Letk be an even integer. We have
dim Mk(Γ) = (k − 1)(g − 1) +Pr i=1 k 2(1 − 1 ei ) +kc 2 , ifs > 2, g + c − 1, ifk = 2, 1, ifk = 0, 0, ifk < 0.
and dim Sk(Γ) = dim Mk(Γ) − c, ifk > 2, g, ifk = 2, 0, ifk ≤ 0.
2.5
Hecke operators on integral weight modular forms
Hecke(1937) introduced a certain ring of operators acting on modular forms. The commutativity of this ring leads to Euler products associated with modular forms. Here we make a brief guide to Hecke operators.
For N ∈ N, if α ∈ GL+2(Z) and Γ0(N ) and α−1Γ0(N )α are commensurable. By [1, sec.1.4], The double coset
Γ0(N )αΓ0(N ) is a finite union of right cosets:
Γ0(N )αΓ0(N ) = h
[
i=1
Γ0(N )αi,
where αi∈ GL+2(Z) and h = [Γ0(N ) : α−1Γ0(N )α]. Then we define a linear operator [Γ0(N )αΓ0(N )] on Mk(Γ0(N ))
by
f |[Γ0(N )αΓ0(N )]k=
X f |αi
In particular, for n ∈ N with gcd(n, N ) = 1 we denote by Tn= n2k−1[Γ0(N ) 1 0 0 n Γ0(N )]k
the Hecke operator of degree n.
Proposition 2.5.1. Let the q-expansion of a modular form f isP
i≥0aiqi, for a primep we have
f |Tp=
X
i≥0
(api+ pk−1ai/p)qi.
We can define an inner product called Peterson inner product on the vector space of cusp forms of weight k on Γ0(N ).
The precise formula is
hf, gi = 1 [SL2(Z) : Γ0(N )] Z Z D ykf (τ )g(τ )dxdy y2 ,
where D is the fundamental domain of Γ0(N ) and we write τ = x + iy for τ ∈ H.With respect to this inner product on
the space of cusp forms, we can show that Tnis self-adjoint if n and N are relatively prime, and thus diagonalizable.
We call f a Hecke eigenform if f is non-vanishing modular form on Γ0(N ) and a simultaneous eigenfunction for all
Hecke operators.
When N = 1, then it can be proved that every Hecke operators commute with each other i.e. TmTn = TnTm. So
we have a nice result from linear algebra that is there is a basis consisting entirely of Hecke eigenforms such that all the Hecke operators are simultaneously diagonalizable. The space of cusps forms of weight k on Γ0(1) is spanned by Hecke
eigenforms. Then the Fourier coefficients aiof f satisfying the following:
1. a16= 0.
2.5. HECKE OPERATORS ON INTEGRAL WEIGHT MODULAR FORMS 7 Thus we may adjust any Hecke eigenform by a constant so that a1= 1. Such Hecke eigenform is called normalized.
And it was shown that the L-function of a Hecke eigenform has a Euler product. Theorem 2.5.2. [1, Theorem1.4.4] If f is a simultaneous eigenform, then
L(s, f ) =Xan ns =
Y
p
(1 − app−s+ pk−1+2s)−1
For example, S12(Γ0(1)) is a one dimensional vector space spanned by a normalized Hecke eigenform ∆(τ ) =
η24(τ ) = P∞
n=1τ (n)q n
, where τ ∈ H and τ (n) are Ramanujan’s tau functiona. Here we obtain the Euler product formula Xτ (n) ns = Y p (1 − τ (p)p−s+ p11−2s)−1 of Ramanujan and Mordell.
But in the cases of N > 1, Sk(Γ0(N )) may not have a basis consisting entirely of simultaneous eigenforms for all
Hecke operators Tn. Here is an example.
Example 2.5.3. Consider the Hecke operator T2acts on S4(Γ0(16)). Then the Jordan form for T2is
0 1 0 0 0 0 0 0 0
Thus, there does not exist a basis whose elements are all simultaneous eigenforms for all Hecke operator. But if f =P∞
n=1anqnis a simultaneous eigenform for all Tn, then f still has the property that Tnf = anf and its
L-function has the Euler product Lf(s) = ∞ X n=1 an ns = Y p|N (1 − app−s)−1 Y p-N (1 − app−s+ pk−1−2s)−1.
The main reason for this is that some of the cusp forms in Sk(Γ0(N )) actually have level smaller than N .
Lemma 2.5.4. [9] Let M , N ∈ N and M dividing N . Then we have St(Γ0(M )) ∈ St(Γ0(N )). And let f ∈ St(Γ0(M )),
then for anyh | (N/M ), the function f (hτ ) ∈ St(Γ0(N )).
Proof. By assumption Let N = kM for some k ∈ Z, and let acN db ∈ Γ0(N ). So a b cN d = a b ckM d ∈ Γ0(M ), thus we have St(Γ0(M )) ∈ St(Γ0(N )). Let γ = a b cN d
. And note that,
hγτ = cNa(hτ ) + hb h (hτ ) + d = a hb cN h d hτ.
By assumption h | (N/M ), we have γ0 = a hb cN h d ∈ Γ0(M ). So f (hγτ ) = f (γ0(hτ )) = (cN τ + d)tf (hτ ). Then f (hτ ) is a cusp form on Γ0(N ).
To define such cusp forms precisely, Let f (τ ) ∈ Sk(Γ0(N )) satisfies f (τ ) = g(hτ ) for some simultaneous eigenform
g(τ ) ∈ Γ0(M ) with M |N and h|(N/M ), then f is called an oldf orm. The space spaned by all oldforms are denoted by
Sold
k (Γ0(N )). And the orthogonal complement of Skold(Γ0(N )) in Sk(Γ0(N )) is call space of newf orms, denoted by
Snew
k (Γ0(N )). In particular, The space Sknew(Γ0(N )) has a basis consisting of simultaneous eigenforms for all Tnwith
gcd(n, N ) = 1.
Now we introduce some theorems we will use: Define the degeneracy map αhas:
αh: Sk(Γ0(M )) −→ Sk(Γ0(N ))
by
αh(f (τ )) = f (hτ ),
where h is the divisors of N/M if N is divisible by M . Proposition 2.5.5. We have a decomposition
Sk(Γ0(N )) = M M |N M d|N/M αd(Snewk (Γ0(M ))). (2.1)
2.6
Modular forms of half integral weight
From now on we let k be an positive odd integer.
To define the modular forms of half integral weight, one may try to make a definition similar to modular forms of integral weight: Let γ be a discrete subgroup of GL+2(R). Assume that f is a holomorphic (or meromorphic) function on H, and it satisfies an appropriate condition at cusps. Then f is a modular form of weight k/2 if
f (γτ ) = (cτ + d)k/2f (τ ), where γ ∈ Γ and τ ∈ H.
Suppose we accept this definition. Then we have a statement:
Proposition 2.6.1. [3] Let Γ0 ⊂ SL2(Z) be any congruence subgroup. Let f be a modular form of weight k/2 satisfies
the above definition. Thenf = 0.
Proof. Γ0is a congruence subgroup of SL2(Z), so we can assume that for some N > 2
Γ(N ) = γ ∈ SL2(Z) : γ ≡ 1 0 0 1 mod N ⊂ Γ0. Let α =N + 1 N −N 1 − N , β = 1 0 −N 1 .
2.6. MODULAR FORMS OF HALF INTEGRAL WEIGHT 9 and we compute αβ =−N 2+ N + 1 N N2− 2N 1 − N . For any nonzero modular form of weight k/2, by the definition we require that
f (ατ ) = (−N τ + (1 − N ))k/2f (τ ) and f (βτ ) = (−N τ + 1)k/2f (τ ). Therefore, f (αβτ ) = (−N (βτ ) + (1 − N ))k/2f (βτ ) = (−N (βτ ) + (1 − N ))k/2(−N τ + 1)k/2f (τ ) = −N τ −N τ + 1 + (1 − N ) k/2 (−N τ + 1)k/2f (τ ) By applying the definition to the matrix αβ directly, we have
f (αβτ ) = ((N2− 2N )τ + (1 − N ))k/2f (τ ) This implies that
((N2− 2N )τ + (1 − N ))k/2=
−N τ
−N τ + 1+ (1 − N ) k/2
(−N τ + 1)k/2. (2.2) When k is even, this equality holds. We may assume that k = 1. Then since the two expressions in the radicals on the right are in lower half plane, the right side is the product of two complex number in the fourth quadrant(we take the branch of the square root having argument in (−π/2, π/2]). But the left side is in the first quadrant, since (N2− 2N )τ + (1 − N ) ∈ H.
Hence (2.2) is wrong by a factor of -1 for k = 1, and also for any odd k.
To see why this definition fails. Note that square root function is multivalued, so our choice of a branch of the square root necessary led to problems. We may handle this group by requiring that our modular forms act on a covering space of GL+2(R), where we allow all branches of the square root simultaneously.
Let an element α =a b c d
∈ GL+2(R) act on H
S
RS{∞} by α(τ ) = (aτ + b)/(cτ + d). Let B denote the set of all couples (α, φ(τ )) formed by an elementa b
c d
of GL+2 and a holomorphic function φ on H such that φ2= t · det(α)−1/2(cτ + d),
where t ∈ T2= {τ ∈ C : |τ | = 1}. Define the law of multiplication by
(α, φ(τ ))(β, ψ(τ )) = (αβ, φ(β(τ ))ψ(τ )), we can make B a group.
Let ξ = (α, φ) ∈ B, we define the action of ξ on CS{∞} to be the same as that of α. Furthermore, for a complex valued function f (τ ) on H and an integer k, we define a function f |[ξ]kon H by
Note that f |[ξη]k = (f |[ξ]k)|[η]k.
Let the function P be the natural projection map defined as: P : B −→ GL+2(R), by
(α, φ) 7→ α. And we denote by L : Γ −→ ∆ the inverse map of P .
Let ∆ be a Fuchsian subgroup of B satisfying the following:
1. P (∆), the projection of ∆ onto SL2(R) is a discrete subgroup, and this projection is one to one.
2. The fundamental domain RP (∆)is of finite measure with respect to the invariant measure y−2dxdy.
3. If −1 ∈ P (∆), then its preimage is (−1, 1).
We call a meromorphic function f (τ ) on H an automorphic form of weight k/2 with respect to ∆ if the following conditions are satisfied:
1. f |[ξ]k = f for all ξ ∈ ∆.
2. f is meromorphic at each cusp of P (∆), where P (∆) is the projection of ∆ on GL+2(R).
We denote by Gk(∆) the vector space of all such f which are holomorphic on H and for which cn = 0 if n < 0, and
further by Sk(∆) the subspace of Gk(∆) consisting of all f for which c0= 0 if r = 0 at every cusp of P (∆).
We should now specialize our discussion to the case where ∆ is obtained from a congruence subgroup of SL2(Z).
Let ∆0(M ) denote the image of Γ0(M ) under L, for every positive integer M divisible by 4. Define an automorphic
factor j(γ, τ ) by j(γ, τ ) = θ(γτ ) θ(τ ) , f or γ ∈ Γ0(4). (2.3) where θ(τ ) = ∞ X n=−∞ e2πin2τ = ∞ X n=−∞ qn2, q = e2πiτ (2.4) and τ is in on H . Then we have by [3, p.148] j(γ, τ )2= −1 d (cτ + d), (2.5) ifa b c d =γ ∈ Γ0(4). In Equation (2.5), −1 d
is the Jacobi symbol. Note that if d is negative, we set −1 d = − −1 |d| .
Here we put γ∗= (γ, j(γ, τ )), then we consider a cusp form f (τ ) satisfying
f (τ )|[γ∗]k= f (γ(τ )) · j(γ, τ )−k (2.6)
for alla b c d
=γ ∈ Γ0(4). Let χ be a Dirichlet character modulo M . We denote by Gk(M, χ)(resp.Sk(M, χ)) the set
of all elements f of Gk(∆0(M )) satisfying
2.7. HECKE OPERATOR TM ON GK/2(M, χ) 11
for all γ ∈ Γ0(M ).
Then the half integral weight modular forms of weight k/2 over Γ0(M ) denoted by Gk(∆0(M )) is the complex
vector space of all such f . If f ∈ Gk(∆0(M )), we see that f (τ + 1) = f (τ ), since
1 1 0 1 , 1 ∈ ∆0(M ). Hence the
Fourier expansion of f has the form f (τ ) =P∞
n=0anqn, where q = e2πiτ.
And then we can define a linear operator on Gk(∆0(M )) for each prime p.
2.7
Hecke operator T
mon G
k/2(M, χ)
Let f (τ ) =∞
X
n=0
af(n)qnin Gk/2(Γ0(M ), χ). Let m be a square of a positive integer, and
α =1 0 0 m
, ξ = (α, m1/4). Suppose we have a disjoint union
∆0(M )ξ∆0(M ) = r [ ν=1 ∆0(M )ξν(disjoint), ΓN(M )αΓ0(M ) = r [ ν=1 Γ0(M )αν.
We define a linear operator on Sk/2(M, χ) by
[∆0(M )ξ∆0(M )]k/2: Gk/2(M, χ) −→ Gk/2(M, χ), by f 7→ f |[∆0(M )ξ∆0(M )]k/2= m(k/4)−1· r X ν=1 χ(aν)f |[ξν]k/2,
which is independent of choice of the representative of ξν.
Definition 2.7.1. The Hecke operator Tmon Gk/2(M, χ) is given by
f |Tm= f |[∆0(M )ξ∆0(M )]k/2= m(k/4)−1· r X ν=1 χ(aν)f |[ξν]k/2, where ξ =1 0 0 m , m1/4 and P (ξν) = aν ∗ ∗ ∗ .
If m is not a square and gcd(N, m) = 1, then [∆0(M )ξ∆0(M )] is a zero operator on Gk/2(M, χ). So we can
consider Tmonly for square m.
In [4] Shimura proved that Tp2: f (τ ) → ∞ X n=0 af(p2n) + χ(p) (−1)λn p pλ−1af(n) + χ(p2)p2λ−1af(n/p2) qn. (2.7) and if n is not divisible by p2, a
2.8
Shimura correspondence
One can define certain liftings of cusp forms of k/2 on Γ0(M ) to cusp forms of weight 2k on Γ0(M0) for a certain M0
depending on M ; these liftings commute with the action of Hecke operators.
Suppose f is a common eigen-function of the operator Tp2for all prime p, and let f |Tp2 = wpf . Define a function F on
H by F (τ ) = ∞ X n=1 Anqn, An∈ C, ∞ X n=1 Ann−s= Y p [1 − ωpp−s+ χ(p)2pk−2−2s]−1,
if k ≥ 5, F is a cusp form of weight k − 1 over Γ0(N ) with character χ2.
In Shimura’s original theorem, the determination of the level M0was a little complicated. However, it was been shown ([5]) that one can always take M0= M/2.
Next let us focus on the space Sk/2+ (4N ) (Kohnen space) of cusp forms of weight k/2 on Γ0(4N ) (N ∈ N is square
free). Recall that the space Sk/2+ (4N ) is the set consisting of elements with the Fourier series of the form X n≥1,(−1)kn≡0,1(4) a(n)qn. Let f = X m≥1 cmqmbe in Sk/2+ (4N, χ). Then define Tk/2,4N,χ(p)(f ) = X m≥1,ε(−1)km≡0,1(4) cp2m + χ(p) ε(−1)km p pk−1cm+ pk−2cm/p2 qm. If we define Pertersson inner product by
hf, gi = 1 [Γ0(4) : Γ]
Z
RΓ
f (τ )g(τ )yk/2−2dxdy (x = Reτ, y = Reτ ),
then Tk/2,4N,χ(p) generate a commutative C-algebra of hermitean operators, and the space Sk/2+ (M, χ) has an orthogonal
basis consisting of common eigenfunctions of all operators Tk/2,4N,χ(p).
Next we define the space of oldforms in Sk0/2(M, χ) to be
X
d|N,d<N
(Sk/2(d, χ) + Sk/2(d, χ)|U (N2/d2))
and the space of newforms Sk/2new(M, χ) to be the orthogonal complement of the space of oldforms in Sk/2+ (M, χ). And since the operator u(f) is an isomorphism between Sk/2+ (M, χ) and Sk/2+ (M ), it is enough to study Sk/2+ (M ), where f is the conductor of χ, and denote by u(f) to be the restriction of U (f) to Sk/2+ (M, χ).
Let HN be the Hecke algebra spanned by the elements Γ0(N ) (a 00 d) Γ0(N ), where a, d > 0, a|d and gcd(d, 2N ) = 1.
Define a linear map R from HN to EndC(S + k/2(4N )) by R Γ0(N ) a 0 0 d Γ0(N ) = a(ad)(k−4)/2 ∆0(4N ) a2 0 0 d2 , (ad)k/2 ∆0(4N ) S+k/2(4N )
2.8. SHIMURA CORRESPONDENCE 13 Then R is a representation of HN[6]. Also, we have a representation
˜ R : HN −→ EndC(Sk−1(N )) defined by ˜ R Γ0(N ) a 0 0 d Γ0(N ) = (ad)k−2 Γ0(N ) a 0 0 d Γ0(N ) k−1
Since R and ˜R are semisimple and in [7] Kohnen showed that
tr(R(ξ), S+k/2(4N )) = tr( ˜R(ξ), Sk−1(N )),
the representations R and ˜R are equivalent.
Recall that for every prime divisor p for N the operator U (p) preserves Snew
2k (N ) and that U (p) = −pk−1wp,2kN on
Snew
2k (N ), where w N
p,2kis Atkin-Lehner involution on S2k(N ) defined by
wp,2kN (f (τ )) = pk(4N τ + pβ)−2kf pτ + α 4N τ + pβ (α, β ∈ Z, p2β − 4N α = p). And there are analogous results for newforms of half integral weight.
One can define an involution on Sk/2(M, χ) as follows. For each prime prime divisor p of N , we define an
”Atkin-Lehner involution” W (p) by W (p) = p a 4N pb , −4 p −k p−k0/4(4N τ + pb)k/2 ! , where a and b are integers with p2b − 4N a = p. In particular, for each prime divisor p of N we put
wp,k/2N = p−k/4+1/2U (p)W (p)
and define S±,pk/2(4N ) as the subspace of S+k/2(4N ) consisting of the forms whose nth Fourier coefficients vanish for (−1)kn p = ∓1; then we set wN p,k/2,χ= u(f) −1wN p,k/2u(f) and S ±,p k/2(4N, χ) = S ±,p k/2(4N )|u(f). The operator wN p,k/2,χis a hermitean involution on S + k/2(4N, χ) whose (±)-eigenspace is S ±,p k/2(4N, χ). In particular,
for each prime divisor of N we have an orthogonal decomposition
Sk/2+ (4N ) = S+,pk/2(4N, χ) ⊕ S−,pk/2(4N, χ). If p does not divide f, then wN
p,k/2,χcoincides with
f
p
p−k/4+1/2U (p)W (p)|Sk/2+ (4N, χ), and S±,pk/2(4N, χ) coincide with the subspace of Sk/2+ (4N, χ) consisting of the forms whose Fourier coefficients vanish for n with(−1)k−1/2p fn= ∓1.
The space Snew
k/2(4N, χ) has an orthogonal basis of common eigenfunctions for all operators Tk/2,4N,χ(p) (p prime,
p does not divide N ), uniquely determined up to multiplication by non-zero complex numbers. These eigenfunctions are also eigenfunctions for the operators U (p2) (p prime, p|N ), the corresponding eigenvalues being ±pk/2−3/2. If f is such
S2knew(N ), uniquely determined up to multiplication with a nonzero complex number, which satisfies Tk0−1,N(p)(F ) =
λpF (resp. U (p2)(F ) = λpF ) for all prime p does not divide N (resp. p|N ). The Fourier coefficients are related as
follows: if f =X
n≥1
anqnand F =
X
n≥1
Anqn, and if D is a fundamental discriminant with ε(−1)kD > 0, then
L(s − k + 1, χ D · )X n≥1 a|D|n2n−s= a(|D|) X n≥1 Ann−s.
Then we define a map ϕD,k/2−1/2,N,χby
X n≥1 bnqn7→ X n≥1 X d|n χ(d) D d dk/2−3/2b n 2 d2|D| qn maps S+k/2(4N, χ) to Sk−1(N ), Sk/2new(4N, χ) to Sk−1new(N ) and for every prime divisor p of N, S
± k/2(4N, χ)T S new k/2(4N, χ) to S±k−1(N )T Snew k−1(N ). It satisfies Tk/2,N,χ(p)ϕD,k/2−1/2,N,χ= ϕD,k/2−1/2,N,χTk−1,N(p)
for all prime p with p - N and U (p2)ϕ
D,k/2−1/2,N,χ = ϕD,k/2−1/2,N,χU (p) for all prime p with p|N . There exist a
linear combination of the ϕD,k/2−1/2,N,χwhich maps Sk/2new(4N, χ)( resp. S ±.p
k/2(4N, χ)T S new
k/2(4N, χ))isomorphically
onto Sk−1new(N ) (resp. S±,pk−1(N )T Snew k−1(N )).
Remark 2.8.1. We see that Snew
k/2(4N, χ) and S new
k−1(N ) are isomorphic, and since strong multiplicity one theorem holds
for Sk−1new(N ), also holds for Snew
k/2(4N, χ).
But it is naturally to ask that does multiplicity one theorem hold for the set of all cusp forms of half-integral weight over Γ0(M )?
The answer is not. Take S13/2(Γ0(4)) for an example. By [?] we know that S13/2(Γ0(4)) isomorphic to S12(Γ0(2))
as modules over the Hecke Algebra. Note that dim S12(Γ0(2)) = 2. And then we compute the matrix representations for
the Hecke operators Tpon S12(Γ0(2)) directly, we see that for p = 3, the Jordan form of the matrix representation is
252 0 0 252
.
It is diagonalizable, so there are two linearly independent eigenfunctions in S12(Γ0(2)) with same eigenvalue. Note that
S12(Γ0(2)) is spanned by {η(τ )24, η(2τ )24}, so S12(Γ0(2)) = S12old(Γ0(2)). That is say multiplicity one theorem does
Chapter 3
Explicit formulas
In this chapter we compute the dimensions of some spaces corresponding to Sr,s(Γ0(3)) which is defined in introduction.
As we mentioned in the introduction, our conjecture holds for r = 1, 5, 7 and fails for r = 11, 13, 17, 19, 23. But there maybe some space of newforms corresponding to Sr,s(Γ0(3)) for large r. From now on, we always let t be a positive
even number.
3.1
Dimension of S
t(Γ
0(2),
1) and S
t(Γ
0(3),
2)
Proposition 3.1.1. Let Snewt (Γ0(2), 1) denote the space of newforms of weight t(t is even) on Γ0(2) that is eigenfunction
forw2with eigenvalues1.Then the dimension of this space is
dim Stnew(Γ0(2), 1) =
(
b3t/8c − bt/3c, if1= 1
t − b3t/8c − bt/3c − bt/4c − 1, if1= −1.
(3.1) Before proving this proposition, we shall prove a lemma first.
Lemma 3.1.2. Let St(Γ0(2), 1) denote the space of cusp forms of weight t(t is even) on Γ0(2) that is eigenfunctions for
w2with eigenvalue1.Then the dimension of this space is
dim(St(Γ0(2), 1)) =
(
b3t/8c + bt/4c − t/2 if1= 1
t/2 − b3t/8c − 1 if1= −1.
(3.2) Proof. Since Γ0(2) is of genus 0, the group Γ+0(2) = Γ0(2) ∪ w2Γ0(2) is of genus 0, where w2 is the Atkin-Lehner
involution on Γ0(2).
And the genus formula says
g = 1 + vol(Γ + 0(2) \ H) 12vol(SL2(Z) \ H) −1 2 r X n=1 (1 − 1 en ) − c 2, (3.3) where vol(Γ \ H) is the volume of the fundamental domain Γ \ H, r is the number of elliptic points, e1, . . . , eris the order
of inequivalent elliptic point, and c is number of cusps.
Since [SL2(Z) : Γ0(2)] = 3, the fundamental domain of Γ+0(2) is equal to 3
2 of the fundamental domain of SL2(Z).
Than we have 0 = 1 +1 8 − r X n=1 (1 − 1 en ) −1 2. 15
(Note that Γ0(2) has two inequivalent cusp are 0, 12, but in Γ +
0(2), 0 and 1
2 are equivalent, since
0 −1 2 0 1 2 = 0). Thus we have r X n=1 (1 − 1 en ) = 5 4.
So we can see that there are one elliptic point of order 2, and one elliptic point of order 4. Then
dim(St(Γ0(2)), +) = b3t/8c + bt/4c − t/2.
Since dim(St(Γ0(2)) = bt/4c − 1, we have
dim(St(Γ0(2), −)) = t/2 − b3t/8c − 1.
Now we can prove the proposition.
Proof of proposition 3.1.1. Let f ∈ St(Γ0(1)) and w2the Atkin-Lehner involution on Γ0(2)
w2= 0 −1 2 0 . Then we have w2(f (τ ) + 2tf (2τ )) = (f (τ ) + 2tf (2τ )) | [w2]t = f (τ ) | [w2]t+ 2tf (2τ ) | [w2]t = det(w2)t/2(−2τ )−tf −1 2τ + 2tdet(w2)t/2(−2τ )−tf −2 2τ = det(w2)t/2(−2τ )−tf 0 −1 1 0 2τ + 2tdet(w2)t/2(−2τ )−tf −1 τ = f (τ ) + 2tf (2τ ).
So the eigenvalue of w2respect to f (τ ) + 2tf (2τ ) is 1. And we also have if f ∈ St(Γ0(1))
(
w2(f (τ )) = 2tf (2τ )
w2(f (2τ )) = 2−tf (τ ).
(3.4) Let {f1(τ ), f2(τ ), . . . , fn(τ )} be a basis for St(Γ0(1)). Then {f1(2τ ), f2(2τ ), . . . , fn(2τ )} is a basis for α2St(Γ0(1)),
where α2is the degeneracy map we defined in chapter 2. Let Q is a subspace of St(Γ0(2), +) defined as
Q = span{f1(τ ) + 2tf1(2τ ), . . . , fn(τ ) + 2nfn(2τ )}.
We claim that
St(Γ0(2), +) = Stnew(Γ0(2), +) ⊕ Q.
By Theorem2.1, we know that
3.1. DIMENSION OF ST(Γ0(2), 1) AND ST(Γ0(3), 2) 17
Suppose we have a g(τ ) ∈ Stnew(Γ0(2), +)T Q. Since Q ⊂ St(Γ0(1))⊕α2St(Γ0(1)) and Stnew(Γ0(2), +) ⊂ Stnew(Γ0(2)),
g ∈ St(Γ0(1)) ⊕ α2St(Γ0(1))T Stnew(Γ0(2)) = {0}.
Next given any f ∈ St(Γ0(2), +). Since St(Γ0(2), +) ⊂ St(Γ0(2)), we can write f as
f = n X i=1 aifi(τ ) + n X i=1 bifi(2τ ) + h(τ ), where h ∈ Snew
t (Γ0(2)). Because of f ∈ St(Γ0(2), +) and newforms are Hecke eigenforms of w2, w2h = h. Then By
w2f = f and Equation(3.4), we have
2t n X i=1 aifi(2τ ) + 2−t n X i=1 bifi(τ ) + h(τ ) = n X i=1 aifi(τ ) + n X i=1 bifi(2τ ) + h(τ ) =⇒ n X i=1 (2tai− bi)fi(2τ ) = n X i=1 (ai− 2−tbi)fi(τ ) =⇒ n X i=1 (ai− 2−tbi)fi(τ ) = 0 =⇒bi= 2tai.
So f ∈ Stnew(Γ0(2), +) ⊕ Q, and then St(Γ0(2), +) = Stnew(Γ0(2), +) ⊕ Q.
Therefore,
dim Stnew(Γ0(2), +) = dim St(Γ0(2), +) − dim St(Γ0(1)
= b3t/8c + bt/4c − t/2 − bt/3c − bt/4c + t/2 = b3t/8c − bt/3c
To compute dim Snew
t (Γ0(2), −), we follow the same process as above but change f (τ ) + 2tf (2τ ) to f (τ ) − 2tf (2τ ).
Similarly, we have formulas for dimension of St(Γ0(3), 2) and Stnew(Γ0(3), 2).
Proposition 3.1.3. Let St(Γ0(3), 2) denote the space of cusp forms of weight t(t is even) on Γ0(3) that is eigenfunctions
forw3with eigenvalue2.Then the dimension of this space is
dim(St(Γ0(3), 2)) =
(
b3t/8c + bt/4c − t/2, if2= 1,
t/2 − b3t/8c − 1, if2= −1.
Moreover, letSnew
t (Γ0(3), 2) denote the space of newforms of weight t(t is even) on Γ0(3) that is eigenfunction for w3
with eigenvalues2.Then the dimension of this space is
dim Snewt (Γ0(3), 2) =
(
b5t/12c − bt/3c, if2= 1
t − b5t/12c − 2bt/4c − 1, if2= −1.
Proposition 3.1.4. Dimension of Snew
t (Γ0(4), 3) is
dim Stnew(Γ0(4), 3) =
(
0 if3= 1
bk/3c − bk/4c if3= −1.
3.2
Dimension of S
t(Γ
0(6),
1,
2)
Proposition 3.2.1. Let Snewt (Γ0(6), 1, 2) denote the space of newforms of weight t(t is even) on Γ0(6) that is
eigen-function forw2,w3with eigenvalues1,2. Then the dimension of this space is
dim Snewt (Γ0(6), 1, 2) = 2bt/4c + bt/3c − b3t/8c − b5t/12c, if1= 1, 2= 1 b5t/12c − b3t/8c, if1= 1, 2= −1, bt/3c − bt/4c + b3t/8c − b5t/12c, if1= −1, 2= 1, bt/4c + b3t/8c + b5t/12c − t + 1, if1= −1, 2= −1.
We prove some lemmas first.
Lemma 3.2.2. Let St(Γ0(6), 1, 2) denote the space of cusp forms of weight t(t is even) on Γ0(6) that is eigenfunctions
forw2,w3with eigenvalues1,2. Then the dimension of this space is
dim St(Γ0(6), 1, 2) 3bt/4c − t/2, if1= 1, 2= 1, t/2 − bt/4c − 1, if1= 1, 2= −1, t/2 − bt/4c − 1, if1= −1, 2= 1, t/2 − bt/4c − 1, if1= −1, 2= −1.
Proof. Γ0(6) is genus 0, so Γ+0(6) = Γ0(6) ∪ w2Γ0(6) ∪ w3Γ0(6) ∪ w6Γ0(6) is genus 0. {0,12, 13, ∞} are inequivalent
cusps of Γ0(6), but in Γ+0(6), we have only one inequivalent cusp, since
w2(0) = 2 −1 6 −2 0 = 1 2.
and also w3(0) = 13, w6(0) = ∞. The fundamental domain of Γ+0(6) is equal to 3 of fundamental domain of SL2(Z),
since the index [SL2(Z, Γ0(6))]=12. And we have
0 = 1 + 3 12− 1 2 r X n=1 (1 − 1 er ) −1 2,
where r is the number of inequivalent elliptic points, and eiis the order of each elliptic point. then r X n=1 (1 − 1 er ) =3 2. so we can see that Γ+0(6) has three elliptic points of order 2.
And then by dimension formula of modular forms, we have dim(St(Γ0(6), +, +)) = 3bt/4c − t/2.
For the others, we first compute the inequivalent cusps and elliptic points by similar method above. Define that 1. (Γ0(6), +, −) = Γ0(6) ∪ w2Γ0(6).
2. (Γ0(6), −, +) = Γ0(6) ∪ w3Γ0(6).
3. (Γ0(6), −, −) = Γ0(6) ∪ w6Γ0(6)
And we see that all of (Γ0(6), +, −) ,(Γ0(6), −, +), and (Γ0(6), −, −) have 2 inequivalent elliptic points of order 2 and 2
inequivalent cusps. So dim(St(Γ0(6), +, +)) = dim(St(Γ0(6), +, −)) = dim(St(Γ0(6), −, −)) = 13(dim(St(Γ0(6)))−
3.2. DIMENSION OF ST(Γ0(6), 1, 2) 19
Lemma 3.2.3. Let f ∈ St(Γ0(2)), then gi(τ ) := f (τ ) + (−1)i3
t
2f (3τ ) ∈ St(Γ0(6)). Moreover the eigenvalue of
Atkin-Lehner operatorw3with respect togiis(−1)i, where i={1, 2}.
Proof. By previous lemma, we see that gi(τ ) ∈ St(Γ0(6)).
Let w3= 3 1 6 3 . Note that w3(f (τ )) = f (τ ) | [w3]t, and f 3τ + 1 6τ + 3 = f (γ3τ ) = (2τ + 3)tf (3τ ), where γ = 1 1 2 3 . So we have w3(f (τ )) = (det(w3))t/2(6τ + 3)−tf 3τ + 1 6τ + 3 = 3t/2f (3τ ), (3.5) and w3(f (3τ )) = (det(w3))t/2(6τ + 3)−tf 3 3τ + 1 6τ + 3 = 3−t/2f (τ ). (3.6) Therefore, w3(gi(τ )) = gi(τ ) |[w3]t= (−1) ig i.
Similarly, we can prove following:
Lemma 3.2.4. Let f ∈ St(Γ0(3)), then hi(τ ) := f (τ ) + (−1)i2
t
2f (2τ ) ∈ St(Γ0(6)). Moreover the eigenvalue of
Atkin-Lehner operatorw2with respect tohiis(−1)i, where i={1, 2}.
And now we can prove the proposition.
Proof of Proposition3.2.1. We prove the case 1= 2= 1, and the others are same. By Equation 2.1 we have an unique
decomposition of St(Γ0(6)) St(Γ0(6)) =St(Γ0(1)) ⊕ α2St(Γ0(1))α3St(Γ0(1))α6St(Γ0(1)) M Stnew(Γ0(2)) ⊕ α3Stnew(Γ0(2)) M Stnew(Γ0(3)) ⊕ α2Stnew(Γ0(3)) M Stnew(Γ0(6)) If f (τ ) ∈ St(Γ0(1)) and let w2= 2 1 6 4
be the atkin-lehner involution on Γ0(6), then we have
w2f (τ ) = 2t/2(6τ + 4)−tf 2τ + 1 6τ + 4 = 2t/2(6τ + 4)−tf1 1 3 4 2τ = 2t/2f (2τ ).
Let w3=
3 1 6 3
be the atkin-lehner involution on Γ0(6), then we have
w3f (τ ) = 3t/2(6τ + 3)−tf 3τ + 1 6τ + 3 = 3t/2(6τ + 3)−tf1 1 2 3 3τ = 3t/2f (3τ ). If f (2τ ) ∈ α2St(Γ0(1)), then we have w2f (2τ ) = 2t/2(6τ + 4)−tf 4τ + 2 6τ + 4 = 2t/2(6τ + 4)−tf 2τ + 1 3τ + 2 = 2t/2(6τ + 4)−tf2 1 3 2 τ = 2−t/2f (τ ), and w3f (2τ ) = 3t/2(6τ + 3)−tf 6τ + 2 6τ + 3 = 3t/2(6τ + 3)−tf1 2 1 3 6τ = 3t/2f (6τ ). If f (3τ ) ∈ α3St(Γ0(1)), then we have w2f (3τ ) = 2t/2(6τ + 4)−tf 6τ + 3 6τ + 4 = 2t/2(6τ + 4)−tf1 3 1 4 6τ = 2t/2f (6τ ), and w3f (3τ ) = 3t/2(6τ + 3)−tf 9τ + 3 6τ + 3 = 3t/2(6τ + 3)−tf3 1 2 1 τ = 3−t/2f (τ ).
3.2. DIMENSION OF ST(Γ0(6), 1, 2) 21 If f (6τ ) ∈ α6St(Γ0(1)), then we have w2f (6τ ) = 2t/2(6τ + 4)−tf 12τ + 6 6τ + 4 = 2t/2(6τ + 4)−tf 6τ + 3 3τ + 2 = 2t/2(6τ + 4)−tf2 3 1 2 3τ = 2−t/2f (3τ ), and w3f (6τ ) = 3t/2(6τ + 3)−tf 18τ + 6 6τ + 3 = 3t/2(6τ + 3)−tf 6τ + 2 2τ + 1 = 3t/2(6τ + 3)−tf3 2 1 1 2τ = 3−t/2f (2τ ).
Let g = af (τ ) + bf (2τ ) + cf (3τ ) + df (6τ ), where a, b, c, d are scalars such that w2g = g, and w3g = g. Then we have
a linear system 2t/2 −1 0 0 0 0 2t/2 −1 3t/2 0 −1 0 0 3t/2 0 −1 a b c d = 0 0 0 0 . (3.7)
Note that this system has only one solution that is (a, b, c, d) = (1, 2t/2, 3t/2, 6t/2). If f (τ ) ∈ Snew
t (Γ2, +), then by
Lemma 3.2.4 and Lemma 3.2.3 we know that
w3f (τ ) = 3t/2f (3τ )
and
w3f (3τ ) = 3−t/2f (τ )
If f (τ ) ∈ Snew
t (Γ3, +), then by Lemma 3.2.4 and Lemma 3.2.3 we know that
w2f (τ ) = 2t/2f (2τ ) and w2f (2τ ) = 2−t/2f (τ ) Now we let V1= span n fi(τ ) + 3t/2fi(3τ ) : fiis basis of St(Γ0(2)) o V2= span n gi(τ ) + 2t/2gi(2τ ) : giis basis of St(Γ0(3)) o V3= span n hi(τ ) + 2t/2hi(2τ ) + 3t/2hi(3τ ) + 6t/2hi(6τ ) : hiis basis of St(Γ0(1)) o .
By considering the decomposition of St(Γ0(6)) and the calculations above we have
St(Γ0(6), +, +) = Stnew(Γ0(6), +, +) ⊕ V1⊕ V2⊕ V3.
So
dim Snewt (Γ0(6), +, +) = dim St(Γ0(6), +, +) − dim V1− dim V2− dim V3 = 2bt/4c + bt/3c − b3t/8c − b5t/12c.
Proposition 3.2.5. Dimension of Snew
t (Γ0(12), 2, 3) is: dim Stnew(Γ0(12), 2, 3) = 0 if2= 1, 3= 1, b5k/12c − bk/3c if2= 1, 3= −1, 0 if2= −1, 3= 1, 2bk/4c − b5k/12c if2= −1, 3= −1,
where3is the eigenvalue of the atkin-lehner involutionw4.
Theorem 3.2.6. [2] Let f =P∞
n=1anqn∈ Mt(Γ0(N ), φ), where φ be a Dirichlet character with conductor c. Let χ be
a primitive Dirichlet character modulom. Then
f ⊗ χ =X
n≥0
χ(n)anqn. (3.8)
andf ⊗ χ ∈ Mt(Γ0(M ), φχ2), where M is the least common multiple of N , cm, m2. Iff is a cusp form, so is f ⊗ χ.
Corollary 3.2.7. Let f ∈ St(Γ0(N )), and χ−4be a primitive Dirichlet character modulo4. Then f ⊗χ−4∈ St(Γ0(16N0), χ2−4),
whereN0 = lcm(16, N )/16. Similarly, Let χ12be Dirichlet primitive character modulo12, then f ⊗χ12∈ St(Γ0(144N0), χ212),
whereN0 = lcm(144, N )/144.
Conjecture 3.2.8. Let r ∈ {1, 5, 7}, and s be a non-negative even integer. Define
Sr,s(Γ0(3)) = {η(24τ )rf (24τ ) : f ∈ Ms(Γ0(3))}. (3.9)
ThenSr,s(Γ0(3)) is an invariant subspace of Sr
2+s(Γ0(576), χ12) under the action of the Hecke algebra. That is for all
primes` 6= 2, 3 and all f ∈ Sr,s(Γ0(3)), we have f | T`2∈ Sr,s(Γ0(3)).
Furthermore, letSnew
t (Γ0(6), 1, 2) denote the space of newforms of weight t on Γ0(6) that are eigenfunctions for w2
andw3with eigenvalues1and2, respectively.
Let alsoSnew
t (Γ0(2), 1) denote the space of newforms of weight t on Γ0(2) that are eigenfunctions for w2with eigenvalue
1.
Then they have
Sr,s(Γ0(3)) ∼= V ⊕ W, (3.10) where V = Sr+2s−1new (Γ0(6), − 2 r , − 3 r ) ⊗ χ12, and W = {Sr+2s−1new (Γ0(2), 2 r ) ⊕ Sr+2s−1new (Γ0(6), 2 r , 3 r ) ⊕ Sr+2s−1new (Γ0(6), 2 r , − 3 r )} ⊗ χ−4.
3.2. DIMENSION OF ST(Γ0(6), 1, 2) 23
For convenience, we let
W1= Sr+2s−1new (Γ0(2), 2r), W2= Sr+2s−1new (Γ0(6), 2r , 3r), W3= Sr+2s−1new (Γ0(6), 2r , − 3r).
And moreover, we also have a conjecture forSr,s(Γ0(2)). Let r ∈ {1, 5, 7}
Sr,s(Γ0(2)) ∼= Sr+2s−1new (Γ0(6), − 2 r , − 3 r ) ⊗ χ12 M Sr+2s−1new (Γ0(12), 3, − 3 r ) ⊗ χ12, (3.11)
Chapter 4
Result
4.1
S
r,s(Γ
0(3))
Example 4.1.1. Consider S1,2(Γ0(3)) = {η(24τ )f (24τ ) : f (τ ) ∈ M2(Γ0(3))} in S2+1/2(Γ0(576), χ12). Let G2(τ )
be the Eisentein series, then f = 3G2(3τ ) − G2(τ )
2 is a modular form on Γ0(3) of weight 2. And by the dimension formula for the modular forms, we know that dim M2(Γ0(3)) = 1, so {f } is a basis. Thus {η(24τ )f (24τ )} is a basis for
S1,2(Γ0(3)), now we apply Hecke operators Tp2on this basis.
For example, let p = 5, then the q-expansion of η(24τ )f (24τ )|T52is
6q + 66q25+ 138q49− 216q73+ 216q97− 138q121− 648q145+ 150q169+ 432q193− 864q217. . . .
and 6η(24τ )f (24τ ) − η(24τ )f (24τ )|T52 = O(q241) means that the initial segment of q-expansions of 6η(24τ )f (24τ )
and η(24τ )f (24τ )|T52agree more than
5
2[SL2(Z) : Γ0(576)]/12 terms, thus η(24τ )f (24τ )|T52 = 6η(24τ )f (24τ ). It is similar for other prime numbers, and we have the following table.
5 7 11 13 17 19 S1,2(Γ0(3)) 6 16 -12 38 -126 -20
On the other hand, since 2r = 1 and 3r = 1, by Proposition 3.1.1 and Proposition 3.2.1 we know that dim V = dim S4(Γ0(6), −, −) = 0 dim W1= dim S4(Γ0(2), +) = 0 dim W2= dim S4(Γ0(6), +, +) = 1 dim W3= dim S4(Γ0(6), +, −) = 0. So V ⊕ W is Snew
4 (Γ0(6), +, +) ⊗ χ−4. We compute the basis of space of newforms by Sage, and the q-expansion of the
basis of Snew
4 (Γ0(6), +, +) ⊗ χ−4is
q + 3q3+ 6q5+ 16q7+ 9q9− 12q11+ 38q13+ 32q14+ 18q15− 126q17− 20q19+ O(q20). And next case are concerned with r = 5, 7 and s = 2.
Example 4.1.2. Sr,2(Γ0(3)) For r = 5, 7.
a. For S5,2(Γ0(3)). Note that,
dim V = dim Snew
5+4−1(Γ0(6), +, +) = 0,
dim W1= dim S5+4−1new (Γ0(2), −) = 0,
dim W2= dim S5+4−1new (Γ0(6), −, −) = 1,
dim W3= dim S5+4−1new (Γ0(6), −, +) = 0.
So from our conjecture
S5,2(Γ0(3)) ∼= S5+4−1new (Γ0(6), −, −) ⊗ χ−4
And we compute eigenvalues for Tp2, p = 5, 7, 11, 13, 17, 19 on S5,2(by Maple)
5 7 11 13 17 19 S5,2(Γ0(3)) -144 1576 -7332 -3802 -6606 -24860
(By Sage.)The normalized newform in Snew
5+4−1(Γ0(6), −, −) is
q + 8q2+ 27q3+ 64q4− 114q5+ 216q6− 1576q7+ 512q8+ . . . ,
then we twist this newform by χ−4:
q − 27q3− 114q5+ 1576q7+ 729q9− 7332q11− 3802q13+ 3078q15− 6606q17− 24860q19+ O(q20).
We can find this normalized newform in S8(Γ0(48)), and its Atkin-Lehner eigenvalues for w2 and w3 are -1 and
1(the eigenvalue of w3change to -1 since we twist by χ−4), respectively.
b. For S5,2(Γ0(3)).
Next we deal with the case r = 7. Note that,
dim V = dim Snew
7+4−1(Γ0(6), +, +) = 0,
dim W1= dim S7+4−1new (Γ0(2), +) = 0,
dim W2= dim S7+4−1new (Γ0(6), +, −) = 1,
dim W3= dim S7+4−1new (Γ0(6), −, +) = 0.
From our conjecture
S7,2(Γ0(3)) ∼= S7+4−1new (Γ0(6), +, −) ⊗ χ−4
And we compute eigenvalues for T2
p, p = 5, 7, 11, 13, 17, 19 on S7,2(Γ0(3))
5 7 11 13 17 19 S7,2(Γ0(3)) 2694 3544 -29580 -44818 -101934 895084
(By Sage.)The normalized newform in Snew
7+4−1(Γ0(6), +, −) is
q − 16q2+ 81q3+ 256q4+ 2694q5− 1296q6− 3544q7− 4096q8+ . . . ,
then we twist this newform by χ−4:
q − 81q3+ 2694q5+ 3544q7+ 6561q9− 29580q11− 44818q13+ 218214q15− 101934q17+ 895084q19+ O(q20).
We can find this normalized newform in S10(Γ0(48)), and its Atkin-Lehner eigenvalues for w2and w3are -1 and
4.1. SR,S(Γ0(3)) 27
For the cases r = 11, 13, 17, 19, 23. Our conjecture never holds by checking the dimensions of Sr,2(Γ0(3)) and V ⊕ W .
By Proposition 3.1.1 and Proposition 3.2.1, we have
dim V ⊕ W = 3 if r = 11, 2 if r = 13, 2 if r = 17, 3 if r = 19, 3 if r = 23. Example 4.1.3. For r = 1, 5, 7 and s = 4. Note that
dim Sr,4(Γ0(3)) = 2. a. For S1,4(Γ0(3)). Note that,
dim V = dim Snew
1+8−1(Γ0(6), −, −) = 1,
dim W1= dim S1+8−1new (Γ0(2), +) = 1,
dim W2= dim S1+8−1new (Γ0(6), +, +) = 0,
dim W3= dim S1+8−1new (Γ0(6), +, −) = 0.
So from our conjecture
S5,2(Γ0(3)) ∼= Snew1+8−1(Γ0(6), −, −) ⊗ χ12
M
S8new(Γ0(2), +) ⊗ χ−4
And we compute eigenvalues for T2
p, p = 5, 7, 11, 13, 17, 19 on S5,2(by Maple)
5 7 11 13 17 19 S1,4 144 1576 7332 -3802 6606 -24860
-210 -1016 -1092 1382 14706 39904 (By Sage.)The normalized newform in Snew
1+8−1(Γ0(6), −, −) is
q + 8q2+ 27q3+ 64q4− 144q5+ 216q6− 1576q7+ 512q8+ . . . ,
then we twist this newform by χ12:
q + 114q5+ 1576q7+ 7332q11− 3802q13+ 6606q17− 24860q19+ O(q20)
We can find this normalized newform in S8(Γ0(144)), and its Atkin-Lehner eigenvalues for w2and w3are -1 and
-1, respectively.
(By Sage.)The normalized newform in Snew
1+8−1(Γ0(2), +) is
q − 8q2+ 12q3+ 64q4− 210q5− 96q6+ 1016q7− 512q8+ . . . .
then we twist this newform by χ−4:
q + 12q3− 210q5− 1016q7− 2043q9− 1092q11+ 1382q13+ 2520q15+ 14706q17+ 39940q19+ O(q20).
b. For S5,4(Γ0(3)). Note that,
dim V = dim Snew
5+8−1(Γ0(6), +, +) = 1,
dim W1= dim S5+8−1new (Γ0(2), −) = 0,
dim W2= dim S5+8−1new (Γ0(6), −, −) = 1,
dim W3= dim S5+8−1new (Γ0(6), −, +) = 0.
From our conjecture
S5,4(Γ0(3)) ∼= S5+8−1new (Γ0(6), +, +) ⊗ χ12
M
S5+8−1new (Γ0(6), +, −) ⊗ χ−4
And we compute eigenvalues for Tp2, p = 5, 7, 11, 13, 17, 19 on S5,4(Γ0(3))
5 7 11 13 17 19 S5,4 -5766 -72464 -408948 -2482858 -5422914 -15166100
3630 -32936 758748 1367558 8290386 10867300 (By Sage.)The normalized newform in Snew
5+8−1(Γ0(6), +, +) is
q − 32q2− 243q3+ 1024q4+ 5766q5+ 7776q6+ 72464q7− 32768q8+ . . . .
then we twist this newform by χ12:
q − 5766q5− 72464q7− 408948q11+ 1367558q13− 5422914q17− 15166100q19+ O(q20)
We can find this normalized newform in S12(Γ0(144)).
(By Sage.)The normalized newform in S5+8−1new (Γ0(6), −, −) is
q + 32q2+ 243q3+ 1024q4+ 3630q5+ 7776q6+ 32936q7+ 32768q8+ . . . , then we twist this newform by χ−4:
q − 243q3+ 3630q5− 32936q7+ 59049q9+ 758748q11− 2482858q13+ 8290386q17+ 10867300q19+ O(q20) We can find this normalized newform in S12(Γ0(144)).
c. For S7,4(Γ0(3)). Note that,
dim V = dim Snew
7+8−1(Γ0(6), −, +) = 1,
dim W1= dim S7+8−1new (Γ0(2), +) = 1,
dim W2= dim S7+8−1new (Γ0(6), +, −) = 0,
dim W3= dim S7+8−1new (Γ0(6), +, +) = 0.
From our conjecture
S7,4(Γ0(3)) ∼= S7+8−1new (Γ0(6), −, +) ⊗ χ12
M
Snew7+8−1(Γ0(2), +) ⊗ χ−4
And we compute eigenvalues for T2
p, p = 5, 7, 11, 13, 17, 19 on S7,4(Γ0(3))
5 7 11 13 17 19 S7,4 -54654 -176336 -1619772 -24028978 60569298 -190034876
4.1. SR,S(Γ0(3)) 29
(By Sage.)The normalized newform in S7+8−1new (Γ0(6), −, +) is
q + 64q2− 729q3+ 4096q4+ 54654q5− 46656q6+ 176336q7+ 262144q8+ . . . .
then we twist this newform by χ12:
q − 54654q5− 176336q7+ 6612420q11− 24028978q13+ 154665054q17− 190034876q19+ O(q20)
We can find this normalized newform in S12(Γ0(144)).
(By Sage.)The normalized newform in Snew
7+8−1(Γ0(2), +) is
q − 64q2− 1836q3+ 4096q4+ 3990q5+ 117504q6− 433432q7− 262144q8+ . . . ,
then we twist this newform by χ−4:
q+1836q3+3990q5+433432q7+1776573q9−1619772q11−10878466q13+60569298q17+243131740q19+O(q20), We can find this normalized newform in S12(Γ0(16)).
Example 4.1.4. For r = 1, 5, 7 and s = 8. dim Sr,6(Γ0(3)) = 3.
a. For S1,8. Note that,
dim V = dim Snew
1+12−1(Γ0(6), −, −) = 1,
dim W1= dim S1+12−1new (Γ0(2), +) = 0,
dim W2= dim S1+12−1new (Γ0(6), +, +) = 1,
dim W3= dim S1+12−1new (Γ0(6), +, −) = 1.
From our conjecture
S1,6(Γ0(3)) ∼= S1+12−1new (Γ0(6), −, +) ⊗ χ12
M {Snew
1+12−1(Γ0(6), +, +) ⊕ S1+12−1new (Γ0(6), +, −)} ⊗ χ−4
And we compute eigenvalues for T2
p, p = 5, 7, 11, 13, 17, 19 on S1,6(Γ0(3))
5 7 11 13 17 19 S1,6(Γ0(3)) -3630 -72464 531420 1332566 5422914 -2901404
5766 50008 -758748 -2482858 -8290386 10867300 -11730 -32936 408948 1367558 -5109678 -15166100 (By Sage.)The normalized newform in S1+12−1new (Γ0(6), −, −) is
q + 32q2+ 243q3+ 1024q4+ 3630q5+ 7776q6+ 32936q7+ 32768q8+ . . . . then we twist this newform by χ12:
q − 3630q5− 32936q7− 758748q11− 2482858q13− 8290386q17+ 10867300q19+ O(q20)
We can find this normalized newform in S12(Γ0(144)).
(By Sage.)The normalized newform inS1+12−1new (Γ0(6), +, +) is
then we twist this newform by χ−4:
q+243q3+5766q5−72464q7+59049q9+408948q11+1367558q13+1401138q15+5422914q17−15166100q19+O(q20)
We can find this normalized newform in S12(Γ0(48)).
(By Sage.)The normalized newform in S1+12−1new (Γ0(6), +, −) is
q − 32q2+ 243q3+ 1024q4− 11730q5− 7776q6− 50008q7− 32768q8+ . . . ,
then we twist this newform by χ−4:
q −11730q5+50008q7+59049q9+531420q11+1332566q13+2850390q15−5109678q17−2901404q19+O(q20)
We can find this normalized newform in S12(Γ0(48)).
b. For S5,6. Note that,
dim V = dim Snew
5+12−1(Γ0(6), +, +) = 1,
dim W1= dim S5+12−1new (Γ0(2), −) = 0,
dim W2= dim S5+12−1new (Γ0(6), −, −) = 1,
dim W3= dim S5+12−1new (Γ0(6), −, +) = 1.
From our conjecture
S5,6(Γ0(3)) ∼= S5+12−1new (Γ0(6), +, +) ⊗ χ12
M {Snew
5+12−1(Γ0(6), −, −) ⊕ S5+12−1new (Γ0(6), −, +)} ⊗ χ−4
And we compute eigenvalues for Tp2, p = 5, 7, 11, 13, 17, 19 on S5,6(Γ0(3))
5 7 11 13 17 19 S5,6(Γ0(3)) -114810 -762104 103451700 -104365834 -3173671566 5895116260
77646 3034528 110255052 56047862 997689762 -2163188180 314490 -2025056 -48011172 285130118 1930104414 -4934015444 (By Sage.)The normalized newform inSnew
5+12−1(Γ0(6), +, +) is
q − 128q2− 2187q3+ 16384q4− 314490q5+ 279936q6+ 2025056q7− 2097152q8+ . . . ,
then we twist this newform by χ12:
q + 314490q5− 2025056q7+ 110255052q11+ 56047862q13+ 1930104414q17− 2163188180q19+ O(q20)
We can find this normalized newform in S16(Γ0(144)).
(By Sage.)The normalized newform inS5+12−1new (Γ0(6), −, −) is
q + 128q2+ 2187q3+ 16384q4+ 77646q5+ 279936q6+ 762104q7+ 2097152q8+ . . . then we twist this newform by χ−4:
q − 2187q3+ 77646q5− 762104q7+ 4782969q9− 48011172q11+ 285130118q13− 169811802q15
4.1. SR,S(Γ0(3)) 31
We can find this normalized newform in S16(Γ0(48)).
(By Sage.)The normalized newform inS5+12−1new (Γ0(6), +, −) is
q + 128q2− 2187q3+ 16384q4− 114810q5− 279936q6− 3034528q7+ 2097152q8+ . . . , then we twist this newform by χ−4:
q + 2187q3− 114810q5+ 3034528q7+ 4782969q9+ 103451700q11− 104365834q13− 251089470q15 + 997689762q17− 4934015444q19+ O(q20).
We can find this normalized newform in S16(Γ0(48)).
c. For S7,6. Note that,
dim V = dim Snew
7+12−1(Γ0(6), −, +) = 1,
dim W1= dim S7+12−1new (Γ0(2), +) = 0,
dim W2= dim S7+12−1new (Γ0(6), +, −) = 1,
dim W3= dim S7+12−1new (Γ0(6), +, +) = 1.
From our conjecture
S5,6(Γ0(3)) ∼= S7+12−1new (Γ0(6), −, +) ⊗ χ12
M {Snew
7+12−1(Γ0(6), +, −) ⊕ S7+12−1new (Γ0(6), +, +)} ⊗ χ−4
And we compute eigenvalues for Tp2, p = 5, 7, 11, 13, 17, 19 on S7,6(Γ0(3))
5 7 11 13 17 19 S7,6(Γ0(3)) 645150 -24959264 -1159304460 -5425661314 -35551782594 -5778498836
199650 -3974432 500068668 2801062862 -5466992958 64354589764 -72186 8640184 125556420 4227195518 32979662226 53889877060 (By Sage.)The normalized newform inSnew
7+12−1(Γ0(6), −, +) is
q + 256q2− 6561q3+ 65536q4− 199650q5− 1679616q6+ 24959264q7+ 16777216q8+ . . . ,
then we twist this newform by χ12:
q + 199650q5− 24959264q7+ 125556420q11+ 4227195518q13− 35551782594q17+ 64354589764q19+ O(q20).
We can find this normalized newform in S18(Γ0(144)).
(By Sage.)The normalized newform inSnew
7+12−1(Γ0(6), +, −) is
q − 256q2+ 6561q3+ 65536q4− 72186q5− 1679616q6− 8640184q7− 16777216q8+ . . .
then we twist this newform by χ−4:
q − 6561q3− 72186q5+ 8640184q7+ 43046721q9− 1159304460q11+ 2801062862q13+ 473612346q15
+ 32979662226q17− 5778498836q19+ O(q20).
(By Sage.)The normalized newform inS7+12−1new (Γ0(6), +, +) is
q − 256q2− 6561q3+ 65536q4+ 645150q5+ 1679616q6+ 3974432q7− 16777216q8+ . . . , then we twist this newform by χ−4:
q + 6561q3+ 645150q5− 3974432q7+ 43046721q9+ 500068668q11− 5425661314q13+ 4232829150q15
− 5466992958q17+ 53889877060q19+ O(q20),
We can find this normalized newform in S18(Γ0(48)).
For the cases r = 11, 13, 17, 19, 23. Our conjecture never holds by checking the dimensions of Sr,2(Γ0(3)) and V ⊕ W .
By Proposition 3.1.1 and Proposition 3.2.1, we have
dim V ⊕ W = 3 if r = 11, 2 if r = 13, 2 if r = 17, 3 if r = 19, 3 if r = 23. Example 4.1.5. For r = 1, 5, 7 and s = 8. dim Sr,8(Γ0(3)) = 3.
a. For S1,8. Note that,
dim V = dim S1+16−1new (Γ0(6), −, −) = 1,
dim W1= dim S1+16−1new (Γ0(2), +) = 1,
dim W2= dim S1+16−1new (Γ0(6), +, +) = 1,
dim W3= dim S1+16−1new (Γ0(6), +, −) = 0.
From our conjecture
S1,8(Γ0(3)) ∼= S1+16−1new (Γ0(6), −, −) ⊗ χ12
M {Snew
1+12−1(Γ0(2), +) ⊕ S1+12−1new (Γ0(6), +, +)} ⊗ χ−4
And we compute eigenvalues for T2
p, p = 5, 7, 11, 13, 17, 19 on S1,6(Γ0(3))
5 7 11 13 17 19 S1,6(Γ0(3)) -314490 -2025056 95889948 56047862 -1355814414 5895116260
90510 -762104 48011172 -59782138 -1930104414 -2163188180 -77646 -56 -110255052 285130118 3173671566 -3783593180 (By Sage.)The normalized newform inS1+12−1new (Γ0(6), −, −) is
q + 32q2+ 243q3+ 1024q4+ 3630q5+ 7776q6+ 32936q7+ 32768q8+ . . . . then we twist this newform by χ12:
q − 3630q5− 32936q7− 758748q11− 2482858q13− 8290386q17+ 10867300q19+ O(q20)
4.1. SR,S(Γ0(3)) 33
(By Sage.)The normalized newform inS1+12−1new (Γ0(6), +, +) is
q − 32q2− 243q3+ 1024q4+ 5766q5+ 7776q6+ 72464q7− 32768q8+ . . . , then we twist this newform by χ−4:
q+243q3+5766q5−72464q7+59049q9+408948q11+1367558q13+1401138q15+5422914q17−15166100q19+O(q20)
We can find this normalized newform in S12(Γ0(48)).
(By Sage.)The normalized newform inS1+12−1new (Γ0(6), +, −) is
q − 32q2+ 243q3+ 1024q4− 11730q5− 7776q6− 50008q7− 32768q8+ . . . , then we twist this newform by χ−4:
q −11730q5+50008q7+59049q9+531420q11+1332566q13+2850390q15−5109678q17−2901404q19+O(q20) We can find this normalized newform in S12(Γ0(48)).
b. For S5,8. Note that,
dim V = dim S5+16−1new (Γ0(6), +, +) = 1,
dim W1= dim S5+16−1new (Γ0(2), −) = 0,
dim W2= dim S5+16−1new (Γ0(6), −, −) = 1,
dim W3= dim S5+16−1new (Γ0(6), −, +) = 1.
From our conjecture
S5,8(Γ0(3)) ∼= S5+16−1new (Γ0(6), +, +) ⊗ χ12
M {Snew
5+16−1(Γ0(6), −, −) ⊕ S5+16−1new (Γ0(6), −, +)} ⊗ χ−4
And we compute eigenvalues for T2
p, p = 5, 7, 11, 13, 17, 19 on S5,6(Γ0(3))
5 7 11 13 17 19 S5,8(Γ0(3)) -114810 -762104 103451700 -104365834 -3173671566 5895116260
77646 3034528 110255052 56047862 997689762 -2163188180 314490 -2025056 -48011172 285130118 1930104414 -4934015444 (By Sage.)The normalized newform inSnew
5+16−1(Γ0(6), +, +) is
q − 128q2− 2187q3+ 16384q4− 314490q5+ 279936q6+ 2025056q7− 2097152q8+ . . . ,
then we twist this newform by χ12:
q + 314490q5− 2025056q7+ 110255052q11+ 56047862q13+ 1930104414q17− 2163188180q19+ O(q20)
We can find this normalized newform in S20(Γ0(144)).
(By Sage.)The normalized newform inSnew
5+16−1(Γ0(6), −, −) is
q + 128q2+ 2187q3+ 16384q4+ 77646q5+ 279936q6+ 762104q7+ 2097152q8+ . . . then we twist this newform by χ−4:
q − 2187q3+ 77646q5− 762104q7+ 4782969q9− 48011172q11+ 285130118q13− 169811802q15
− 3173671566q17+ 5895116260q19+ O(q20)
We can find this normalized newform in S20(Γ0(48)).
(By Sage.)The normalized newform inS5+16−1new (Γ0(6), +, −) is
q + 128q2− 2187q3+ 16384q4− 114810q5− 279936q6− 3034528q7+ 2097152q8+ . . . ,
then we twist this newform by χ−4:
q + 2187q3− 114810q5+ 3034528q7+ 4782969q9+ 103451700q11− 104365834q13− 251089470q15
+ 997689762q17− 4934015444q19+ O(q20).
We can find this normalized newform in S20(Γ0(48)).
c. For S7,8. Note that,
dim V = dim Snew
7+16−1(Γ0(6), −, +) = 1,
dim W1= dim S7+16−1new (Γ0(2), +) = 0,
dim W2= dim S7+16−1new (Γ0(6), +, −) = 1,
dim W3= dim S7+16−1new (Γ0(6), +, +) = 1.
From our conjecture
S7,8(Γ0(3)) ∼= S7+16−1new (Γ0(6), −, +) ⊗ χ12
M
{S7+16−1new (Γ0(6), +, −) ⊕ S7+16−1new (Γ0(6), +, +)} ⊗ χ−4
And we compute eigenvalues for Tp2, p = 5, 7, 11, 13, 17, 19 on S7,8(Γ0(3))
5 7 11 13 17 19 S7,8(Γ0(3)) 645150 -24959264 -1159304460 -5425661314 -35551782594 -5778498836
199650 -3974432 500068668 2801062862 -5466992958 64354589764 -72186 8640184 125556420 4227195518 32979662226 53889877060 (By Sage.)The normalized newform inSnew
7+16−1(Γ0(6), −, +) is
q + 256q2− 6561q3+ 65536q4− 199650q5− 1679616q6+ 24959264q7+ 16777216q8+ . . . ,
then we twist this newform by χ12:
q + 199650q5− 24959264q7+ 125556420q11+ 4227195518q13− 35551782594q17+ 64354589764q19+ O(q20). We can find this normalized newform in S22(Γ0(144)).
(By Sage.)The normalized newform inSnew
7+16−1(Γ0(6), +, −) is
4.2. SR,S(Γ0(2)) 35
then we twist this newform by χ−4:
q − 6561q3− 72186q5+ 8640184q7+ 43046721q9− 1159304460q11+ 2801062862q13+ 473612346q15
+ 32979662226q17− 5778498836q19+ O(q20).
We can find this normalized newform in S22(Γ0(48)).
(By Sage.)The normalized newform inSnew
7+16−1(Γ0(6), +, +) is
q − 256q2− 6561q3+ 65536q4+ 645150q5+ 1679616q6+ 3974432q7− 16777216q8+ . . . , then we twist this newform by χ−4:
q + 6561q3+ 645150q5− 3974432q7+ 43046721q9+ 500068668q11− 5425661314q13+ 4232829150q15
− 5466992958q17+ 53889877060q19+ O(q20), We can find this normalized newform in S22(Γ0(48)).
For the cases r = 11, 13, 17, 19, 23. Our conjecture never holds by checking the dimensions of Sr,2(Γ0(3)) and V ⊕ W .
By Proposition 3.1.1 and Proposition 3.2.1, we have
dim V ⊕ W = 3 if r = 11, 2 if r = 13, 2 if r = 17, 3 if r = 19, 3 if r = 23.
4.2
S
r,s(Γ
0(2))
We take ( V = Snew t (Γ0(6), − 2r , − 3r) ⊗ χ12, U = Snew t (Γ0(6), 3, − 3r) ⊗ χ12, for convenience.Example 4.2.1. For r = 1, 5, 7 and s = 2. a. For S1,2(Γ0(2)).
Note that,
(
dim V = dim Snew
4 (Γ0(6), −, −) = 0,
dim U = dim S4new(Γ0(12), −, −) = 1.
So from our conjecture
S1,2(Γ0(2)) ∼= Snew1+2−1(Γ0(12), −, −) ⊗ χ12
And we compute eigenvalues for T2
p, p = 5, 7, 11, 13, 17, 19 on S5,2(by Maple)
5 7 11 13 17 19 S1,2(Γ0(2)) 18 -8 36 -10 -18 100
