Algorithms for Computing with Modular Forms by William Stein

By William Stein

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0 (N ) or Γ1 (N ). 1. [Find γ] Using the extended Euclidean algorithm, find γ ∈ SL2 (Z) such that γ(∞) = α. If α = ∞ set γ ← 1; otherwise, write α = a/b, find c, d such that ad − bc = 1, and set γ ← ac db . 2. [Generic Conjugate Matrix] Compute the following matrix in M2 (Z[x]): δ(x) ← γ 1 x −1 γ . 0 1 Note that δ(x) matrix whose entries are constant or linear in x. 3. [Solve] The congruence conditions that define Γ give rise to four linear congruence conditions on x. Use techniques from elementary number theory to find the smallest simultaneous positive solution h to these four equations.

Make sure this is right for N ≤ 5. 1. We have dim S2 (Γ1 (N )) = g1 (N ). 1. If k ≥ 3, let a = (k − 1)(g1 (N ) − 1) + k − 1 · c1 (N ). 2 Then for N ≥ 3,   a + 1/2 dim Sk (Γ1 (N )) = a + k/3   a if N = 4 and 2 k, if N = 3, otherwise. The dimension of the Eisenstein subspace is as follows: dim Ek (Γ1 (N )) = c1 (N ) c1 (N ) − 1 if k = 2, if k = 2. 3. 3. 2. Since Mk = Sk ⊕ Ek , the formulas above also give a formula for the dimension of Mk . 3 dim S2 (Γ1 (N )) 0 0 1 231 6112 28921 109893 299792001 dim S3 (Γ1 (N )) 0 2 5 530 12416 58920 221444 599792000 dim S4 (Γ1 (N )) 0 5 10 830 18721 88920 332996 899792000 dim S24 (Γ1 (N )) 2 65 110 6830 144821 688920 2564036 6899792000 Modular Forms with Character Fix a Dirichlet character ε modulo N , and let c be the conductor of ε (we do not assume that ε is primitive).

9. People also often write that f has “nebentypus character” ε. I rarely hear anyone actually say nebentypus, and it’s somewhat redundant, so I will simply omit it in this book. , forms that vanish at all cusps (elements of Q ∪ {∞}), and Ek (N, ε) is the subspace of Eisenstein series, which is the unique subspace of Mk (N, ε) that is invariant under all Hecke operators and is such that Mk (N, ε) = Sk (N, ε) ⊕ Ek (N, ε). The space Ek (N, ε) can also be defined as the space spanned by all Eisenstein series of weight k and level N , as defined below.

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