
# q-expansion of newform 637.2.e.m, 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 = 637
weight = 2
poly_data = [4, 16, 64, 28, 58, 18, 41, 7, 8, -1, 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], 1], [[0, 1, 0, 0, 0, 0, 0, 0, 0, 0], 1], [[30518, 2376, 9416, 84549, 6908, 880, -5913, -220, 176, -364], 118350], [[-22604, -98928, -392048, -39747, -287624, -36640, -87336, 9160, -7328, -983], 118350], [[-348592, -132894, -526654, 25194, -386377, -49220, -109053, 12305, -9844, -1159], 118350], [[-180842, -41094, -162854, -129531, -119477, -15220, -3978, 3805, -3044, 916], 59175], [[-3988, -20316, -46906, 39066, -152578, 31195, -34617, 26720, -9541, 2249], 78900], [[-2692, -957744, -258154, -871206, -272902, -637445, -107253, -121720, 14531, -15259], 236700], [[532, 1362924, 255184, 981276, 92092, 719570, 96813, 145570, -21776, 18139], 118350], [[-136, 50058, 11648, 53832, 2993, 39580, 3726, 9041, -1552, 1058], 4734]]

hecke_ring_character_values = [[248, [0, 0, 0, 0, 0, 0, 0, -1, 0, 0]], [197, [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]]]
aps_data = [[0, 1, 0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 0, 0, -1, 0, 0, 1, 2, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, -2, -2, 0, -1, -1, 0, 1, 1, 1], [0, 2, 0, 0, 0, 0, -1, 2, -1, -1], [0, 0, 0, 0, 0, 0, 0, -2, 0, -1], [-1, 0, -2, 0, -1, 0, 0, 0, 0, 0], [-2, 0, 0, 0, 1, 2, 0, 2, -2, -1], [0, -2, 0, 0, 0, 0, -2, 0, 0, -1], [-4, 0, -2, 2, -3, 1, 0, 0, 0, 0], [0, 0, 2, 2, -1, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, -4], [-3, 4, 4, 2, 2, 0, -2, 3, 0, -2], [2, 0, 0, 1, 0, 0, -1, -2, 0, 0], [0, 0, 0, 0, 0, 0, 4, -3, 0, -2], [-4, 2, 2, 3, 1, 3, -3, 4, -3, -1], [2, 0, -4, 1, -1, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, -1, -2, 0, 1, 0], [0, -2, 0, 0, 0, 0, 2, 0, -1, -2], [-2, 0, 2, 0, 1, 3, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, -2, 0, -2, 3], [2, 0, 2, 2, -1, 1, 0, 0, 0, 0], [-2, 0, 0, 0, 2, -3, 0, 2, 3, -2], [0, -8, 0, 0, 0, 0, -2, 0, 2, 3], [0, -2, 0, 0, 0, 0, 0, -12, 1, 2], [0, -6, -6, 2, -1, 0, -2, 0, 0, 1], [1, 0, 2, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, -2, -2, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 2, -4, -3, -2], [-6, -4, -4, 2, -6, 1, -2, 6, -1, 6], [4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 8, 0, 0, 0, 0, 2, 4, -1, -2], [10, 2, 2, -3, 5, -1, 3, -10, 1, -5], [-7, 2, 2, -2, 1, 1, 2, 7, -1, -1], [0, -2, 0, 0, 0, 0, -3, -4, 3, 1], [-6, 0, -8, -3, 1, -2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -4, -1, -2, 2], [-6, 0, 0, 0, 3, -2, 0, 6, 2, -3], [-3, 0, -6, 2, -1, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, -2, -6, -1, 0], [-4, -2, -2, -2, -1, 4, 2, 4, -4, 1], [0, 0, 2, 0, 5, -1, 0, 0, 0, 0], [12, 4, 4, 2, -5, 0, -2, -12, 0, 5], [0, 0, -2, -6, -3, -1, 0, 0, 0, 0], [-2, 0, 10, -1, 8, -1, 0, 0, 0, 0], [-2, 4, 4, -4, 1, 2, 4, 2, -2, -1], [0, -2, 0, 0, 0, 0, 0, 8, 2, -7], [0, -4, 0, 0, 0, 0, 2, 1, 2, 4], [8, 0, 2, -1, 4, 5, 0, 0, 0, 0], [-2, 6, 6, -4, 5, 0, 4, 2, 0, -5], [14, 0, 2, -4, 1, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -4, 6, -1, 4], [-8, 0, 0, 0, 5, 2, 0, 8, -2, -5], [-7, 6, 6, -2, 3, 3, 2, 7, -3, -3], [0, -2, 0, 0, 0, 0, 1, -4, 1, 1], [5, 8, 8, -4, 2, -2, 4, -5, 2, -2], [2, 0, 2, 4, -7, -3, 0, 0, 0, 0], [-10, 6, 6, 0, 0, -3, 0, 10, 3, 0], [8, 0, -6, -4, 5, -9, 0, 0, 0, 0], [0, 0, 6, 5, -4, 3, 0, 0, 0, 0], [0, 4, 4, -4, 5, 4, 4, 0, -4, -5], [0, -4, 0, 0, 0, 0, 2, 0, 3, 4], [0, 0, 0, 0, 0, 0, -6, 8, -5, -2], [0, 0, 0, 0, 0, 0, 4, 6, 4, -9], [-7, 0, -2, -2, -3, 6, 0, 0, 0, 0], [-16, 4, 4, 0, 5, -2, 0, 16, 2, -5], [2, 0, 2, -2, 3, 5, 0, 0, 0, 0], [8, 2, 2, 4, -1, -4, -4, -8, 4, 1], [0, -4, 0, 0, 0, 0, -8, 2, 4, 7], [14, -2, -2, 4, 6, 3, -4, -14, -3, -6], [0, 2, 0, 0, 0, 0, -6, 7, -5, 7], [0, 0, 10, -2, 11, 1, 0, 0, 0, 0], [0, 4, 0, 0, 0, 0, 2, 0, -2, 3], [-5, -6, -6, 4, 7, 1, -4, 5, -1, -7], [0, 4, 0, 0, 0, 0, -6, 4, -5, 0], [0, -6, 0, 0, 0, 0, 0, -4, 4, 1], [-10, -4, -4, -6, 2, -1, 6, 10, 1, -2], [6, 0, 0, 4, -4, 2, 0, 0, 0, 0], [-10, 0, 0, 0, 0, 0, 0, 0, 0, 0], [12, 0, 0, -2, 5, 0, 2, -12, 0, -5], [7, 0, 4, -4, 2, -1, 0, 0, 0, 0], [0, 6, 0, 0, 0, 0, 2, -20, 3, -4], [0, 10, 0, 0, 0, 0, 2, -22, 1, -2], [-8, 0, -4, 4, -6, -4, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 0, 0, -2, -9], [8, 0, 2, -2, 11, 3, 0, 0, 0, 0], [2, 0, 2, -4, -3, 3, 0, 0, 0, 0], [0, -6, 0, 0, 0, 0, -2, 8, -1, 0], [10, -8, -8, 7, -4, -6, -7, -10, 6, 4], [12, 12, 12, 5, 2, 6, -5, -12, -6, -2], [10, 0, -4, -8, 6, -4, 0, 0, 0, 0], [0, -8, 0, 0, 0, 0, 0, 6, 4, 3], [-2, 0, 10, -2, -1, 7, 0, 0, 0, 0], [0, -6, 0, 0, 0, 0, -2, 16, 4, 1], [-8, 4, 4, 10, -4, 3, -10, 8, -3, 4], [0, -8, 0, 0, 0, 0, -4, 14, -6, 3], [0, -4, 0, 0, 0, 0, 0, 12, -3, 2], [-6, 0, -14, 4, -7, -5, 0, 0, 0, 0], [2, -8, -8, -4, 0, -7, 4, -2, 7, 0], [-8, -10, -10, -4, 8, 3, 4, 8, -3, -8], [0, 8, 0, 0, 0, 0, 6, 17, -4, 2], [-2, -4, -4, -6, -1, -4, 6, 2, 4, 1], [2, -2, -2, 6, -17, 4, -6, -2, -4, 17], [10, 0, -8, 5, 1, -6, 0, 0, 0, 0], [0, 8, 0, 0, 0, 0, -12, 12, -1, -2], [-6, 6, 6, 2, -2, 11, -2, 6, -11, 2], [-13, 0, 6, -8, -5, 0, 0, 0, 0, 0], [0, -6, 0, 0, 0, 0, -8, -14, 7, 0], [8, -4, -4, 4, -6, -7, -4, -8, 7, 6], [4, 0, 2, -4, -7, 7, 0, 0, 0, 0], [16, -12, -12, 6, -5, -4, -6, -16, 4, 5], [-6, 0, -6, -4, 9, -1, 0, 0, 0, 0], [0, 0, 0, 6, -18, 7, -6, 0, -7, 18], [10, 0, 8, -7, -3, -6, 0, 0, 0, 0], [0, 2, 2, -10, -6, -1, 10, 0, 1, 6], [0, -4, 0, 0, 0, 0, -6, -4, -1, -10], [-4, 0, 16, 4, 4, 8, 0, 0, 0, 0], [-12, 2, 2, -2, -9, 2, 2, 12, -2, 9], [-12, 0, -8, 10, -10, -2, 0, 0, 0, 0], [0, 8, 0, 0, 0, 0, 8, 1, 4, -14], [-28, -12, -12, -2, -3, -4, 2, 28, 4, 3], [0, 16, 0, 0, 0, 0, -5, -6, -4, 0], [2, 0, -4, -4, 10, 2, 0, 0, 0, 0], [0, -8, 0, 0, 0, 0, 2, 8, 7, 2], [0, 8, 0, 0, 0, 0, 0, -18, -4, 1], [-6, 0, -14, -2, -5, -1, 0, 0, 0, 0], [0, -10, 0, 0, 0, 0, 2, 14, -4, -1], [-10, -4, -4, 8, -1, -2, -8, 10, 2, 1], [6, 0, -10, -11, 4, -11, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -10, -6, 0, -7], [11, 0, 0, -2, 2, -7, 0, 0, 0, 0], [0, -8, 0, 0, 0, 0, 4, 4, 3, -6], [2, 0, 4, -4, -2, 6, 0, 0, 0, 0], [-4, 6, 6, 4, 3, 0, -4, 4, 0, -3], [26, 10, 10, 1, -3, -3, -1, -26, 3, 3], [-8, 0, 8, 10, 6, 2, 0, 0, 0, 0], [3, 2, 2, -4, -3, -5, 4, -3, 5, 3], [26, 0, -6, 0, 5, -5, 0, 0, 0, 0], [0, 4, 0, 0, 0, 0, 2, 2, 7, -8], [12, 8, 8, -4, 1, -10, 4, -12, 10, -1], [28, 0, -4, -1, 5, -2, 0, 0, 0, 0], [1, -12, -12, 2, -10, -6, -2, -1, 6, 10], [-20, 0, -2, 7, 2, 7, 0, 0, 0, 0], [0, 0, 12, -2, 12, 4, 0, 0, 0, 0], [-3, -8, -8, -12, -8, -6, 12, 3, 6, 8], [0, 6, 0, 0, 0, 0, -4, -20, -3, 8], [0, 8, 0, 0, 0, 0, 16, -18, 2, -15], [0, -14, 0, 0, 0, 0, 2, -16, -2, 17], [0, 0, -12, -14, -4, -4, 0, 0, 0, 0], [16, 0, -2, 4, 7, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 16, -14, 4, -11], [-2, 2, 2, 0, -4, -13, 0, 2, 13, 4], [-16, 0, -2, 2, -3, -5, 0, 0, 0, 0], [0, 16, 0, 0, 0, 0, 8, -10, 4, -13], [0, 2, 0, 0, 0, 0, 2, 22, 4, 7], [9, 0, 0, 12, -8, -1, 0, 0, 0, 0], [-12, 10, 10, 6, -5, 6, -6, 12, -6, 5], [0, 0, 0, 0, 0, 0, -7, -16, 2, 8], [-17, 0, -4, 10, -12, -1, 0, 0, 0, 0], [8, 0, 0, 11, -5, -8, 0, 0, 0, 0], [0, 18, 0, 0, 0, 0, 2, -6, 5, -8], [6, 8, 8, 0, -10, 3, 0, -6, -3, 10], [0, 6, 6, 1, 5, 7, -1, 0, -7, -5], [8, 4, 4, -4, 7, 0, 4, -8, 0, -7], [3, 8, 8, 14, -4, -2, -14, -3, 2, 4]]
