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Let $f \in S_2(\Gamma(N))$ be a weight 2 cusp form of full level $N > 5$. It has a Fourier expansion:

$$ f(z) = a_1 q^{1/N} + a_2q^{2/N} + \cdots. $$

Now let $S = \left(\begin{array}{cc} 0 & -1 \newline 1 & 0 \end{array}\right)$. It is not in $\Gamma(N)$, so $f|_2S \neq f$. (Here $|_2$ denotes the usual weight 2 'slash' operator.)

Nevertheless $f|_2S \in S_2(\Gamma(N))$, since $\Gamma(N)$ is normal in $SL_2(\mathbb{Z})$. Hence it too has a Fourier expansion.

What is it's Fourier expansion (in terms of the $a_i$s)?

Whatever it is, it must be such that, acting by $S$ again I get back to the $a_i$s. This suggests to me that the answer might be "complex conjugate, with possibly a minus one at the front", but I can't seem to rigorously convince myself of this (if indeed it is true), nor understand where this possible minus one comes from.

Any hints welcome.

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The map $f \mapsto f|_2 S$ is clearly a $\mathbb{C}$-linear map from $S_2(\Gamma(N), \mathbb{C})$ to itself and thus cannot possibly be complex conj'n on the coefficients (with or without a minus sign). Complex conj'n -- more formally the map $f(z) \mapsto f(-\bar{z})$ -- corresponds to the action of $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$. – David Loeffler Oct 30 '11 at 14:42
up vote 4 down vote accepted

If you conjugate $\Gamma(N)$ by $\begin{pmatrix} N & 0 \\ 0 & 1 \end{pmatrix}$ so it gets sent to a subgroup intermediate between $\Gamma_1(N^2)$ and $\Gamma_0(N^2)$ -- which corresponds to replacing $q^{1/N}$ by $q$ -- then this is just the Atkin-Lehner involution, up to some easily evaluated constant fudge factor.

This of course raises the question of explicitly computing the A-L involution on the relevant subspace of $S_2(\Gamma_1(N^2))$. I don't know a nice way of doing this, but there are certainly plenty of nasty ones.

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Thanks for your quick response David. There is a 'nice' way that I know of (depending on one's interpretation of 'nice') when $N$ is a power of a prime $q$, $f$ is a newform, and $a_q(f) \neq 0$; it involves the Gauss sum of a Dirichlet Character, and is in a paper of Atkin and Li; (alas, in some of my examples $a_q(f) = 0$, so I may have to use some nastier methods). – Barinder Banwait Oct 30 '11 at 20:58

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