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Let $N>0$, and let $Q \parallel N$ satisfy $\gcd(Q,N/Q)=1$.

Let $x,y,z,t \in \Z$ be such that the determinant of \[ W_Q=\left( \begin{matrix} Qx & y \\ Nz & Qt\end{matrix} \right) \] is $Q$. Then $W_Q$ normalizes the group $\Gamma_0(N)$ and so induces an involution $w_Q$ of the modular curve $X_0(N)$ which does not depend on the choice of $x,y,z$ and $t$.

The Atkin-Lehner involution $W_Q$ commutes with the Hecke operators $T_p$ for $p \nmid Q$. A cusp form $f$ in $S_k(\Gamma_0(N))$ which is an eigenform for all $T_p$ with $p \nmid N$ is also an eigenform for $W_Q$, with eigenvalue $\pm 1$.

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  • Review status: reviewed
  • Last edited by John Voight on 2020-10-29 17:16:45
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