
# q-expansion of newform 370.2.n.f, downloaded from the LMFDB on 25 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 = 370
weight = 2
poly_data = [4, -8, 8, -24, 20, 20, -8, 16, 4, -8, 2, -2, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 1], [[748, 48, -48, 144, -146, -94, 22, -18, 28, -4, 1, -1], 460], [[612, 148, -148, 444, -454, -286, 64, -44, 94, -20, 5, -5], 460], [[-12, 12, -1370, -50, 110, -414, 264, 166, -103, 43, -43, 12], 460], [[-142, -154, 154, 1148, 524, 246, 560, -109, -201, 9, -31, 31], 230], [[12, -12, 1140, 50, -110, 414, -264, -166, 103, -43, 43, -12], 230], [[-32, 32, -1982, -118, 278, -1058, 658, 412, -267, 107, -107, 32], 460], [[-420, -484, 484, 3424, 1614, 806, 1208, -244, -566, 48, -81, 81], 460], [[-12, 32, -38, 86, 74, -12, 46, 4, -23, 5, -5, 3], 20], [[476, 524, -524, -3856, -1774, -846, -1734, 344, 666, -36, 101, -101], 460], [[-1060, 1060, -1294, 2790, 2510, -92, 1700, 250, -941, 195, -195, 117], 460], [[612, -612, 732, -1636, -1424, 184, -906, -94, 469, -100, 100, -60], 230]]

hecke_ring_character_values = [[297, [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], [261, [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0]]]
aps_data = [[0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, -1, -1, 0, 1, 0, 0], [0, 1, 0, 0, 0, 0, -1, -1, 0, 0, 0, -1], [0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0], [-3, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -2, 1, 1, 0, -2, 0, 0], [0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0], [0, -2, 4, 0, 0, 0, 0, 0, 2, 0, 4, 2], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0], [-2, -6, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, -2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, -4, -3, -2, -1, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 0, -3, 0, 0, -2], [0, 0, 0, 0, -4, 0, 0, -1, 0, -1, 0, 0], [0, 0, 0, 0, 2, 0, 0, -2, 0, 1, 0, 0], [0, 0, 0, 3, 0, -1, 0, 0, 0, 0, 0, 0], [5, 0, 0, 0, 0, 0, 0, 0, -5, 0, -1, 2], [0, 4, 2, 0, 0, 0, 0, 0, 4, 0, 2, -4], [0, 0, 0, -9, -9, 3, 1, 1, 0, 3, 0, 0], [0, 2, -2, 0, 0, 0, 0, 0, -6, 0, -2, -2], [0, 0, 0, 0, 9, 0, 0, 1, 0, 2, 0, 0], [0, 1, 5, 0, 0, 0, 0, 0, 1, 0, 5, -1], [0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0], [-7, 0, 0, 0, 0, 0, 0, 0, 7, 0, -1, -3], [0, 0, 0, 0, -6, 0, 0, -6, 0, -4, 0, 0], [-6, 1, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 6, 0, 0, 4, 0, 1, 0, 0], [0, 0, 0, -8, -8, -5, -9, -9, 0, -5, 0, 0], [8, 0, 0, 0, 0, 0, 0, 0, -8, 0, -8, 0], [0, 0, 0, -2, 0, -4, -6, 0, 0, 0, 0, 0], [0, 0, 0, -14, 0, -4, -4, 0, 0, 0, 0, 0], [8, 0, 0, 0, 0, 0, 0, 0, -8, 0, 0, -4], [0, 0, 0, 0, -1, 0, 0, -5, 0, -2, 0, 0], [-3, 0, 0, 0, 0, 0, 0, 0, 3, 0, 1, 2], [4, -8, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 5, 4, 0, 0, 0, 0, 0, -1, 0, 4, -5], [0, 0, 0, 13, 0, 1, 2, 0, 0, 0, 0, 0], [0, 0, 0, 6, 0, -5, -3, 0, 0, 0, 0, 0], [0, 0, 0, 14, 14, -2, -2, -2, 0, -2, 0, 0], [0, 0, 0, 15, 0, -3, -3, 0, 0, 0, 0, 0], [8, -4, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 2, 0, 0, 0, 0, 0, 18, 0, 2, 1], [7, 3, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -9, 0, 0, 5, 0, -6, 0, 0], [0, 0, 0, 5, 0, -5, -2, 0, 0, 0, 0, 0], [9, 1, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4, -10, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 10, 0, 0, -6, 0, -1, 0, 0], [0, 0, 0, 1, 1, 8, -6, -6, 0, 8, 0, 0], [0, -7, -6, 0, 0, 0, 0, 0, 10, 0, -6, 7], [0, 0, 0, 0, -3, 0, 0, -1, 0, 8, 0, 0], [7, 0, 0, 0, 0, 0, 0, 0, -7, 0, 9, 7], [0, -6, 4, 0, 0, 0, 0, 0, 2, 0, 4, 6], [-10, 8, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -9, 0, -8, 1, 0, 0, 0, 0, 0], [0, 0, 0, -4, -4, -1, 0, 0, 0, -1, 0, 0], [12, 3, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0], [10, 0, 0, 0, 0, 0, 0, 0, -10, 0, -9, -4], [0, 0, 0, -6, -6, -4, -5, -5, 0, -4, 0, 0], [-8, 0, 0, 0, 0, 0, 0, 0, 8, 0, 13, 5], [0, 0, 0, 4, 4, -7, 11, 11, 0, -7, 0, 0], [0, 0, 0, -2, -2, 14, 7, 7, 0, 14, 0, 0], [0, 0, 0, 0, -22, 0, 0, -11, 0, -5, 0, 0], [-11, 0, 0, 0, 0, 0, 0, 0, 11, 0, 10, -1], [0, 0, 0, 10, 0, 2, 10, 0, 0, 0, 0, 0], [0, 0, 0, -13, 0, 5, -2, 0, 0, 0, 0, 0], [7, 0, 0, 0, 0, 0, 0, 0, -7, 0, -15, -4], [0, 0, 0, 8, 8, 6, 14, 14, 0, 6, 0, 0], [0, 0, 0, 0, 5, 0, 0, -5, 0, -5, 0, 0], [-2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, -12], [0, 0, 0, 9, 0, 4, 13, 0, 0, 0, 0, 0], [-9, 11, -10, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 14, 8, 8, 0, 14, 0, 0], [0, 0, 0, -5, -5, 5, -8, -8, 0, 5, 0, 0], [3, 0, 0, 0, 0, 0, 0, 0, -3, 0, -7, 0], [0, 0, 0, 4, 4, 1, 4, 4, 0, 1, 0, 0], [0, -10, 8, 0, 0, 0, 0, 0, 10, 0, 8, 10], [0, 0, 0, 0, -4, 0, 0, -13, 0, -2, 0, 0], [15, -9, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0], [11, 0, 0, 0, 0, 0, 0, 0, -11, 0, -2, 10], [-11, 0, 0, 0, 0, 0, 0, 0, 11, 0, -3, 0], [-12, 3, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, -4, 0, 0, 0, 0, 0, -31, 0, -4, -1], [0, 0, 0, 0, -2, 0, 0, 4, 0, 0, 0, 0], [0, -10, 9, 0, 0, 0, 0, 0, -6, 0, 9, 10], [0, 0, 0, 0, 5, 0, 0, 6, 0, -6, 0, 0], [0, 7, 11, 0, 0, 0, 0, 0, 2, 0, 11, -7], [0, 0, 0, 14, 14, -14, -8, -8, 0, -14, 0, 0], [12, 0, 0, 0, 0, 0, 0, 0, -12, 0, 6, -4], [0, 0, 0, -4, -4, -2, -14, -14, 0, -2, 0, 0], [0, 0, 0, 0, -18, 0, 0, 5, 0, 1, 0, 0], [-23, 0, 0, 0, 0, 0, 0, 0, 23, 0, -4, 1], [0, 0, 0, 0, -12, 0, 0, 12, 0, 2, 0, 0], [7, 10, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, -2, 0, 0, 0, 0, 0, -24, 0, -2, 2], [0, 0, 0, 4, 0, -7, -4, 0, 0, 0, 0, 0], [-4, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, -3], [0, -5, 7, 0, 0, 0, 0, 0, -4, 0, 7, 5], [0, 0, 0, -23, -23, -2, 4, 4, 0, -2, 0, 0], [-16, -8, -10, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -7, 0, 0, -2, 0, 4, 0, 0], [0, 0, 0, -7, 0, -1, 10, 0, 0, 0, 0, 0], [0, 0, 0, 0, -5, 0, 0, 1, 0, 9, 0, 0], [9, 0, -10, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-4, 0, 0, 0, 0, 0, 0, 0, 4, 0, -20, 2], [0, 0, 0, -7, 0, 6, -13, 0, 0, 0, 0, 0], [0, 0, 0, 25, 25, 6, 0, 0, 0, 6, 0, 0], [0, 0, 0, 0, 17, 0, 0, 19, 0, 14, 0, 0], [0, 12, -5, 0, 0, 0, 0, 0, 10, 0, -5, -12], [4, 0, 0, 0, 0, 0, 0, 0, -4, 0, -7, -19], [0, 0, 0, -6, 0, -9, 0, 0, 0, 0, 0, 0], [0, 0, 0, -24, 0, -4, -18, 0, 0, 0, 0, 0], [0, 0, 0, -6, 0, 16, -2, 0, 0, 0, 0, 0], [-22, -4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15, -12], [8, 0, 0, 0, 0, 0, 0, 0, -8, 0, -5, 7], [0, 0, 0, 0, -19, 0, 0, -4, 0, -2, 0, 0], [0, 0, 0, -10, -10, 0, 16, 16, 0, 0, 0, 0], [0, 0, 0, -11, -11, -7, -6, -6, 0, -7, 0, 0], [0, -8, 6, 0, 0, 0, 0, 0, 10, 0, 6, 8], [0, 9, -16, 0, 0, 0, 0, 0, 2, 0, -16, -9], [0, 0, 0, -1, -1, -16, -11, -11, 0, -16, 0, 0], [0, 0, 0, 0, 18, 0, 0, 0, 0, -6, 0, 0], [0, 0, 0, -12, 0, 21, 9, 0, 0, 0, 0, 0], [-7, 0, 0, 0, 0, 0, 0, 0, 7, 0, -7, -18], [-10, 0, 0, 0, 0, 0, 0, 0, 10, 0, 12, 6], [-26, 10, -24, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-19, 0, 0, 0, 0, 0, 0, 0, 19, 0, 0, 1], [0, 0, 0, 20, 20, -2, -10, -10, 0, -2, 0, 0], [0, 0, 0, 16, 16, 18, 18, 18, 0, 18, 0, 0], [5, 6, 19, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 9, 16, 16, 0, 9, 0, 0], [13, -15, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -16, 0, 16, 11, 0, 0, 0, 0, 0], [-15, 0, 0, 0, 0, 0, 0, 0, 15, 0, -9, 13], [-21, -7, -9, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -26, -26, -8, 1, 1, 0, -8, 0, 0], [0, 0, 0, 0, 18, 0, 0, 21, 0, 3, 0, 0], [0, 0, 0, -19, 0, -3, -12, 0, 0, 0, 0, 0], [0, -6, 22, 0, 0, 0, 0, 0, 13, 0, 22, 6], [0, 14, 2, 0, 0, 0, 0, 0, -10, 0, 2, -14], [0, 16, -24, 0, 0, 0, 0, 0, -12, 0, -24, -16], [0, 0, 0, -8, 0, 1, 8, 0, 0, 0, 0, 0], [0, 0, 0, 7, 7, 9, -3, -3, 0, 9, 0, 0], [-22, 0, 0, 0, 0, 0, 0, 0, 22, 0, 4, 9], [16, 0, 0, 0, 0, 0, 0, 0, -16, 0, -13, -6], [0, 0, 0, 15, 0, 21, 24, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 9, 0, -14, 0, 0], [28, -8, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 18, 0, -2, 10, 0, 0, 0, 0, 0], [0, 0, 0, 0, 17, 0, 0, 6, 0, 17, 0, 0], [0, -13, 11, 0, 0, 0, 0, 0, 3, 0, 11, 13], [0, 0, 0, 7, 7, -10, 8, 8, 0, -10, 0, 0], [0, 0, 0, 0, -12, 0, 0, 2, 0, 4, 0, 0], [0, 0, 0, 13, 13, -5, -5, -5, 0, -5, 0, 0], [4, 26, -7, 0, 0, 0, 0, 0, 0, 0, 0, 0], [6, -8, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, 22, 0, 0, 0, 0, 0, -5, 0, 22, 2], [0, 0, 0, -16, 0, 4, 4, 0, 0, 0, 0, 0], [-16, 0, 0, 0, 0, 0, 0, 0, 16, 0, -2, 13], [0, 0, 0, 2, 0, -7, -21, 0, 0, 0, 0, 0], [0, 0, 0, -3, 0, -8, 21, 0, 0, 0, 0, 0], [0, 0, 0, 10, 10, 22, 10, 10, 0, 22, 0, 0], [-9, 0, 0, 0, 0, 0, 0, 0, 9, 0, 31, 4], [0, 0, 0, 28, 0, -12, 4, 0, 0, 0, 0, 0], [0, 0, 0, 14, 0, -10, -6, 0, 0, 0, 0, 0], [-10, -8, -5, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 23, 0, 5, 12, 0, 0, 0, 0, 0]]
