Function: mfatkininit
Section: modular_forms
C-Name: mfatkininit
Prototype: GLp
Help: mfatkininit(mf,Q): initializes data necessary for working
 with Atkin--Lehner operators W_Q, for now only the function mfatkin.
 The result is a 4-component vector [mfB, MC, C, mf] where mfB is either
 0 or the possibly different modular form space to which F|W_Q will belong
 (this does not depend on F in mf); MC is the matrix of W_Q on the basis of mf
 multiplied by a normalizing constant C.
Doc: given a modular form space with parameters $N,k,\chi$ and a
 primitive divisor $Q$ of the level $N$, initializes data necessary for
 working with the Atkin--Lehner operator $W_{Q}$, for now only the function
 \kbd{mfatkin}. We write $\chi \sim \chi_{Q} \chi_{N/Q}$ where
 the two characters are primitive with (coprime) conductors dividing
 $Q$ and $N/Q$ respectively. For $F\in M_{k}(\Gamma_{0}(N),\chi)$,
 the form $F | W_{Q}$ still has level $N$ and weight $k$ but its
 Nebentypus may no longer be $\chi$: it becomes
 $\overline{\chi_{Q}} \chi_{N/Q})$
 if $k$ is integral and $\overline{\chi_{Q}} \chi_{N/Q})(4Q/\cdot)$ if not.

 The result is a technical 4-component vector \kbd{[mfB, MC, C, mf]}, where

 \item \kbd{mfB} encodes the modular form space to which
 $F|W_{Q}$ belongs when $F \in M_{k}(\Gamma_{0}(N), \chi)$: an \kbd{mfinit}
 corresponding to a new Nebentypus or the integer $0$ when the character does
 not change. This does not depend on $F$.

 \item \kbd{MC} is the matrix of $W_{Q}$ on the bases of \kbd{mf} and \kbd{mfB}
 multiplied by a normalizing constant $C(k,\chi,Q)$. This matrix has polmod
 coefficients in $\Q(\chi)$.

 \item \kbd{C} is the complex constant $C(k,\chi,Q)$. For $k$
 integral, let $A(k,\chi, Q) = Q^{\varepsilon}/g(\chi_{Q})$, where
 $\varepsilon = 0$ for $k$ even and $1/2$ for $k$ odd and
 where $g(\chi_{Q})$ is the Gauss sum attached to $\chi_{Q}$). (A similar, more
 complicated, definition holds in half-integral weight depending on the parity
 of $k - 1/2$.)  Then if $M$ denotes the matrix of $W_{Q}$ on the bases
 of \kbd{mf} and \kbd{mfB}, $A \cdot M$ has coefficients in $\Q(\chi)$.
 If $A$ is rational, we let $C = 1$ and $C = A$ as a floating point complex
 number otherwise, and finally $\kbd{MC} := M \cdot C$.

 \bprog
 ? mf=mfinit([32,4],0); [mfB,MC,C]=mfatkininit(mf,32); MC
 %1 =
 [5/16 11/2  55/8]

 [ 1/8    0  -5/4]

 [1/32 -1/4 11/16]

 ? C
 %2 = 1
 ? mf=mfinit([32,4,8],0); [mfB,MC,C]=mfatkininit(mf,32); MC
 %3 =
 [  1/8 -7/4]

 [-1/16 -1/8]
 ? C
 %4 = 0.35355339059327376220042218105242451964
 ? algdep(C,2)   \\ C = 1/sqrt(8)
 %5 = 8*x^2 - 1
 @eprog
