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Also [CO77] does not contain any proofs that their claimed formulas are correct, but instead say only that “Les formules qui les donnent sont connues de beaucoup de gens et il existe plusieurs m´ethodes permettant de les obtenir (th´eor`eme 54CHAPTER 4. 1. 2 Track this down the analogue of Moebius inversion for µ and give a quick presentation on it. 3 Implement in your favorite computer language an algorithm to compute dim Sk (Γ0 (N )). Chapter 5 Linear Algebra This chapter is about exact matrix algebra with over the rational numbers and cyclotomic fields.

Loop over prime divisors] Set i ← i + 1. If i > n, return t. Otherwise set p ← pi and e ← ei . (a) (b) (c) (d) (e) (f) If p ≡ 3 (mod 4), return 0. If p = 2 and e > 1, return 0. If p = 2 and e = 1, go to Step 3. 4. Compute ω = a(p−1)/4 . Using the Chinese Remainder Theorem to find x ∈ Z/N Z such that x ≡ a (mod p) and x ≡ 1 (mod N/pe ). 3. MODULAR FORMS WITH CHARACTER (g) (h) (i) (j) 53 r−1 Set x ← xp . Set s ← ε(x). If s = 1, set t ← 2t and go to Step 3. If s = −1, set t ← −2t and go to Step 3.

This theorem reduces the problem of computing Sk (Γ0 (N )) to that of computing Sk (Γ0 (M ))new for divisors M of N , a fact that will be central later in 48CHAPTER 4. COMPUTING DIMENSIONS OF SPACES OF MODULAR FORMS this book. Atkin and Lehner also prove that one can completely determine Sk (Γ0 (M ))new just from the information of how the Hecke operators act on it (their “multiplicity one” theory). D. thesis under A. Ogg (see [Li75]). If N | N | N , then the maps αd from Mk (Γ0 (N )) to Mk (Γ0 (N )) factor through Mk (Γ0 (N )).

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