By Larry Joel Goldstein

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**Example text**

3] by writing out Tn (f ) explicitly and using that 0≤b

Choose a = b = 0, since k ≡ 0 (mod 12). Then g1 = ΔF62 = q − 1032q 2 + 245196q 3 + 10965568q 4 + 60177390q 5 − · · · and g2 = Δ2 = q 2 − 48q 3 + 1080q 4 − 15040q 5 + · · · . We let f2 = g2 and f1 = g1 + 1032g2 = q + 195660q 3 + 12080128q 4 + 44656110q 5 − · · · . 22 2. 23. When k = 36, the Miller basis including f0 is 6218175600q 4 + 15281788354560q 5 + · · · , f0 = 1 + f1 = q+ 57093088q 4 + 37927345230q 5 + · · · , 194184q 4 + 7442432q 5 + · · · , q2 + f2 = q3 − f3 = 2484q 5 + · · · . 24. The SAGE command victor miller basis computes the Miller basis to any desired precision for a given k.

Proof. Suppose γ ∈ SL2 (Z). Since γ induces an automorphism of Z2 , Xn · γ = {δγ : δ ∈ Xn } is also in bijection with the subgroups of Z2 of index n. For each element δγ ∈ Xn · γ, there is σ ∈ SL2 (Z) such that σδγ ∈ Xn (the element σ transforms δγ to Hermite normal form), and the set of elements σδγ is thus equal to Xn . Thus f [σδγ]k = Tn,k (f ) = σδγ∈Xn f [δγ]k = Tn,k (f )[γ]k . δ∈Xn A ﬁnite sum of meromorphic function is meromorphic, so Tn,k (f ) is weakly modular. If f is holomorphic on h, then each f [δ]k is holomorphic on h for δ ∈ Xn .

### Analytic Number Theory by Larry Joel Goldstein

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