
# q-expansion of newform 2304.3.b.f, downloaded from the LMFDB on 19 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 = 2304
weight = 3
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 = [[1279, [-1, 0]], [2053, [-1, 0]], [1793, [1, 0]]]
aps_data = [[0, 0], [0, 0], [0, 3], [0, 0], [0, 0], [0, -5], [30, 0], [0, 0], [0, 0], [0, 21], [0, 0], [0, -35], [18, 0], [0, 0], [0, 0], [0, -45], [0, 0], [0, 11], [0, 0], [0, 0], [110, 0], [0, 0], [0, 0], [-78, 0], [130, 0], [0, 99], [0, 0], [0, 0], [0, 91], [30, 0], [0, 0], [0, 0], [210, 0], [0, 0], [0, 51], [0, 0], [0, -85], [0, 0], [0, 0], [0, 165], [0, 0], [0, -19], [0, 0], [-190, 0], [0, 195], [0, 0], [0, 0], [0, 0], [0, 0], [0, 221], [210, 0], [0, 0], [418, 0], [0, 0], [510, 0], [0, 0], [0, 69], [0, 0], [0, -115], [-462, 0], [0, 0], [0, -285], [0, 0], [0, 0], [-50, 0], [0, -75], [0, 0], [-350, 0], [0, 0], [0, 299], [-450, 0], [0, 0], [0, 0], [0, -275], [0, 0], [0, 0], [0, -189], [0, -325], [798, 0], [782, 0], [0, 0], [0, 29], [0, 0], [290, 0], [0, 0], [0, 0], [702, 0], [-850, 0], [0, 261], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, -459], [-558, 0], [0, 0], [0, -341], [0, 0], [0, 165], [0, 0], [-462, 0], [0, 0], [-1150, 0], [0, 0], [-930, 0], [0, 0], [1102, 0], [0, 0], [0, -35], [210, 0], [0, 0], [0, 0], [-1218, 0], [0, 0], [0, 0], [0, -315], [0, 0], [0, 589], [770, 0], [0, 675], [0, 0], [0, 0], [0, -651], [0, -259], [0, 0], [0, 0], [0, -725], [0, 0], [0, 0], [0, 0], [0, -595], [-78, 0], [962, 0], [0, 195], [0, 0], [0, -555], [-1518, 0], [0, 0], [0, -429], [0, 0], [0, 0], [0, -629], [0, 0], [0, 205], [1650, 0], [0, 0], [0, 0], [0, -805], [-738, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-258, 0], [430, 0], [0, 741], [0, 0], [-1230, 0], [0, 0], [0, 0], [-1890, 0], [0, 0], [0, 0], [0, 925], [-1118, 0], [0, -45], [0, 0], [0, 779], [0, 0], [2030, 0], [0, 0], [-1998, 0], [0, 0], [0, -861], [0, 0], [0, 731], [0, 0], [0, 0], [0, 1085], [1170, 0], [0, 0], [0, -141], [0, 235], [0, 0], [-658, 0], [0, 0], [2050, 0], [0, 0], [0, 0], [0, -1131], [0, 0], [-1710, 0], [98, 0], [0, -245], [-1410, 0], [0, 0], [0, 1221], [0, 0], [0, -1075], [-1598, 0], [0, 0], [0, -1035], [0, 0], [0, 0], [2322, 0], [0, 0], [-2590, 0], [0, 51], [0, 0], [0, 0], [0, 0], [2542, 0], [0, 0], [-1122, 0], [0, 0], [0, 1365], [0, -931], [0, 0], [318, 0], [0, 0], [0, 0], [0, -371], [2610, 0], [0, 0], [0, 0], [0, 0], [0, 1435], [0, 0], [0, 0], [1938, 0], [0, 0], [0, 0], [1378, 0], [0, 1395], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, -901], [990, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 715], [3198, 0], [0, 0], [3182, 0], [0, -1275], [0, 0], [0, 1421], [0, 0], [0, -285], [1870, 0], [0, 0], [0, 0], [0, -1219], [0, -1045], [-3330, 0], [0, 0], [0, 741], [-2958, 0], [0, 0], [0, 1155], [0, 59], [0, 0], [590, 0], [0, 0], [2530, 0], [0, 0], [0, 0], [0, 1739], [-1298, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 61], [0, 0], [0, 0], [610, 0], [0, -1485], [0, 0], [2622, 0], [0, 549], [0, 0], [3570, 0], [0, 0], [0, 1595], [0, 1749], [0, 0], [0, 915], [0, 0], [0, 0], [-3410, 0], [0, -315], [0, 0]]
