mfatkineigenvalues

Function: mfatkineigenvalues
Section: modular_forms
C-Name: mfatkineigenvalues
Prototype: GLp
Help: mfatkineigenvalues(mf,Q): given a modular form space mf
 and a primitive divisor Q of the level of mf, outputs the corresponding
 Atkin-Lehner eigenvalues on the new space, grouped by orbit.
Doc: Given a modular form space \kbd{mf} of integral weight $k$ and a primitive
 divisor $Q$ of the level $N$ of \kbd{mf}, outputs the Atkin--Lehner
 eigenvalues of $w_Q$ on the new space, grouped by orbit. If the Nebentypus
 $\chi$ of \kbd{mf} is a
 (trivial or) quadratic character defined modulo $N/Q$, the result is rounded
 and the eigenvalues are $\pm i^k$.
 \bprog
 ? mf = mfinit([35,2],0); mffields(mf)
 %1 = [y, y^2 - y - 4] \\ two orbits, dimension 1 and 2
 ? mfatkineigenvalues(mf,5)
 %2 = [[1], [-1, -1]]
 ? mf = mfinit([12,7,Mod(3,4)],0);
 ? mfatkineigenvalues(mf,3)
 %4 = [[I, -I, -I, I, I, -I]]  \\ one orbit
 @eprog
 To obtain the eigenvalues on a larger space than the new space,
 e.g., the full space, you can directly call \kbd{[mfB,M,C]=mfatkininit} and
 compute the eigenvalues as the roots of the characteristic polynomial of
 $M/C$, by dividing the roots of \kbd{charpoly(M)} by $C$. Note that the
 characteristic polynomial is computed exactly since $M$ has coefficients in
 $\Q(\chi)$, whereas $C$ may be given by a complex number. If the coefficients
 of the characteristic polynomial are polmods modulo $T$ they must be embedded
 to $\C$ first using \kbd{subst(lift(), t, exp(2*I*Pi/n))}, when $T$ is
 \kbd{poliscyclo(n)}; note that $T = \kbd{mf.mod}$.