 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2003/iii/paper-24/3/ii/solution

Choose

b = (a d - 1) / N \in Z

, and take the matrix

γ = (a N b d) \in S L_{2} (Z) .

w = τ - d / N

Nw + d = N τ

, while its numerator is

a w + b = a τ - 1/ N

. Consequently

γ w = \frac{a}{N} - \frac{1}{N ^{2} τ} .

Applying the weight-

k

transformation law of the cusp form at

w \in H

f (\frac{a}{N} - \frac{1}{N ^{2} τ}) = (N τ)^{k} f (τ - \frac{d}{N}) .

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