
# q-expansion of newform 800.4.c.f, downloaded from the LMFDB on 24 April 2024.

# We generate the q-expansion using the Hecke eigenvalues a_p at the primes.
# Each a_p is given as a linear combination
# of the following basis for the coefficient ring.

def make_data():

    from sage.all import prod, floor, prime_powers, gcd, QQ, primes_first_n, next_prime, RR

    def discrete_log(elts, gens, mod):
        # algorithm 2.2, page 16 of https://arxiv.org/abs/0903.2785
        def table_gens(gens, mod):
            T = [1]
            n = len(gens)
            r = [None]*n
            s = [None]*n
            for i in range(n):
                beta = gens[i]
                r[i] = 1
                N = len(T)
                while beta not in T:
                    for Tj in T[:N]:
                        T.append((beta*Tj) % mod)
                    beta = (beta*gens[i]) % mod
                    r[i] += 1
                s[i] = T.index(beta)
            return T, r, s
        T, r, s = table_gens(gens, mod)
        n = len(gens)
        N = [ prod(r[:j]) for j in range(n) ]
        Z = lambda s: [ (floor(s/N[j]) % r[j]) for j in range(n)]
        return [Z(T.index(elt % mod)) for elt in elts]
    def extend_multiplicatively(an):
        for pp in prime_powers(len(an)-1):
            for k in range(1, (len(an) - 1)//pp + 1):
                if gcd(k, pp) == 1:
                    an[pp*k] = an[pp]*an[k]
    from sage.all import PolynomialRing, NumberField, ZZ
    R = PolynomialRing(QQ, "x")
    f = R(poly_data)
    K = NumberField(f, "a")
    betas = [K([c/ZZ(den) for c in num]) for num, den in basis_data]
    convert_elt_to_field = lambda elt: sum(c*beta for c, beta in zip(elt, betas))
    # convert aps to K elements
    primes = primes_first_n(len(aps_data))
    good_primes = [p for p in primes if not p.divides(level)]
    aps = map(convert_elt_to_field, aps_data)
    if not hecke_ring_character_values:
        # trivial character
        char_values = dict(zip(good_primes, [1]*len(good_primes)))
    else:
        gens = [elt[0] for elt in hecke_ring_character_values]
        gens_values = [convert_elt_to_field(elt[1]) for elt in hecke_ring_character_values]
        char_values = dict([(
            p,prod(g**k for g, k in zip(gens_values, elt)))
            for p, elt in zip(good_primes, discrete_log(good_primes, gens, level))
            ])
    an_list_bound = next_prime(primes[-1])
    an = [0]*an_list_bound
    an[1] = 1
    
    from sage.all import PowerSeriesRing
    PS = PowerSeriesRing(K, "q")
    for p, ap in zip(primes, aps):
        if p.divides(level):
            euler_factor = [1, -ap]
        else:
            euler_factor = [1, -ap, p**(weight - 1) * char_values[p]]
        k = RR(an_list_bound).log(p).floor() + 1
        foo = (1/PS(euler_factor)).padded_list(k)
        for i in range(1, k):
            an[p**i] = foo[i]
    extend_multiplicatively(an)
    return PS(an)
level = 800
weight = 4
poly_data = [1, 0, 1]

# The entries in the following list give a basis for the
# coefficient ring in terms of a root of the defining polynomial above.
# Each line consists of the coefficients of the numerator, and a denominator.
basis_data  = [[[1, 0], 1], [[0, 2], 1]]

hecke_ring_character_values = [[351, [1, 0]], [101, [1, 0]], [577, [-1, 0]]]
aps_data = [[0, 0], [0, 1], [0, 0], [0, 3], [60, 0], [0, -25], [0, -15], [-40, 0], [0, -89], [-166, 0], [20, 0], [0, 5], [-250, 0], [0, -71], [0, 107], [0, -245], [800, 0], [250, 0], [0, -387], [100, 0], [0, 115], [1320, 0], [0, -491], [-874, 0], [0, -155], [-1498, 0], [0, -701], [0, 597], [-650, 0], [0, 755], [0, 623], [2660, 0], [0, 1385], [560, 0], [2350, 0], [580, 0], [0, -655], [0, -931], [0, 363], [0, -1625], [1120, 0], [-2842, 0], [3180, 0], [0, 2335], [0, -1495], [4240, 0], [-4060, 0], [0, 2811], [0, 777], [-1134, 0], [0, 855], [-4440, 0], [-850, 0], [-660, 0], [0, -3795], [0, -381], [150, 0], [-6580, 0], [0, 2265], [6950, 0], [0, 1941], [0, -685], [0, 2053], [2220, 0], [0, 4715], [0, -3235], [900, 0], [0, 265], [0, -207], [-8614, 0], [0, 1135], [-8080, 0], [0, 1187], [0, -905], [-8120, 0], [0, 5891], [4350, 0], [0, -3735], [11698, 0], [3650, 0], [1120, 0], [4850, 0], [-12580, 0], [0, -6565], [-8560, 0], [0, 2129], [-2550, 0], [0, -3355], [-14482, 0], [0, -81], [0, -7987], [10760, 0], [0, -4633], [2860, 0], [-7160, 0], [0, 699], [-7446, 0], [-16438, 0], [0, 3661], [10878, 0], [0, 8057], [0, 1845], [0, 1281], [6050, 0], [8260, 0], [0, -8435], [0, -483], [0, -13145], [11640, 0], [-25450, 0], [0, 8347], [0, -7945], [0, -615], [10840, 0], [-14060, 0], [-17650, 0], [0, -13679], [0, -3393], [0, 4515], [15600, 0], [16850, 0], [0, 3995], [0, 9345], [0, -9591], [-23380, 0], [11850, 0], [-25646, 0], [30280, 0], [0, 8723], [0, 8375], [36560, 0], [0, 15071], [11860, 0], [0, 18505], [-11718, 0], [-4706, 0], [0, 14335], [0, 10217], [0, 1965], [4854, 0], [-13140, 0], [22050, 0], [0, -7289], [0, -18527], [6150, 0], [8200, 0], [0, 21495], [0, 16065], [-15440, 0], [0, -23469], [0, -15615], [25550, 0], [0, -2159], [0, 883], [0, 20953], [25140, 0], [-32920, 0], [-10150, 0], [0, 14265], [9678, 0], [0, 18493], [0, 1675], [0, 21887], [8740, 0], [0, -24155], [0, -1141], [-31580, 0], [0, -1395]]
