charpolys_degree2.py

from degree2.all import CuspFormsDegree2


def compute_charpolys(k):
    '''
    Assuming the charstestic polynomial of T(2) on S_k(Sp(4, Z))
    is square free over Q,
    return a pair of polynomials (pl1, pl2) s.t.
    pl1 is the characteristic polynomial of T(2)
    pl2 is the characteristic polynomial of T_1(4).
    '''
    R = PolynomialRing(QQ, names="x")
    x = R.gens()[0]
    if k % 2 == 0:
        prec = max((k // 10) * 2, 4)
    else:
        prec = max(((k - 5)//10) * 2, 12)
    S = CuspFormsDegree2(k, prec=prec)
    pols = []
    s_charpoly = S.hecke_charpoly(2)
    if any(i > 1 for _, i in s_charpoly.factor()):
        raise RuntimeError("charpoly is not square free.")

    for pl1, _ in s_charpoly.factor():
        if pl1.degree() == 1:
            eg = - pl1[0][0]
        else:
            eg = NumberField(pl1, name="a").gens()[0]
        f = S.eigenform_with_eigenvalue_t2(eg)
        eg_t4 = f.hecke_eigenvalue(4)
        eg_t1_4 = (eg**2 - eg_t4 - 2**(2 * k - 4) - 10 * 2**(2 * k - 6))/2
        if eg_t1_4 in QQ:
            pl2 = x - eg_t1_4
        else:
            pl2 = eg_t1_4.charpoly()
        pols.append((pl1, pl2))
    return (mul(a for a, _ in pols), mul(b for _, b in pols))