
# q-expansion of newform 1900.2.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 = 1900
weight = 2
poly_data = [1, 0, 18, 0, 9, 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, 0, 0, 0, 0], 1], [[0, 1, 0, 0, 0, 0], 1], [[1, 0, 5, 0, 1, 0], 3], [[10, 0, 8, 0, 1, 0], 3], [[0, 13, 0, 8, 0, 1], 3], [[0, 41, 0, 19, 0, 2], 3]]

hecke_ring_character_values = [[951, [1, 0, 0, 0, 0, 0]], [77, [-1, 0, 0, 0, 0, 0]], [401, [1, 0, 0, 0, 0, 0]]]
aps_data = [[0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, -1], [0, 0, 0, 0, 0, 0], [0, 2, 0, 0, -1, -1], [0, 0, 0, -1, 0, 0], [0, 1, 0, 0, 1, -1], [0, -2, 0, 0, -1, -1], [-1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 5, 0], [-2, 0, 2, 1, 0, 0], [-1, 0, 3, -1, 0, 0], [0, 0, 0, 0, 4, 1], [0, 0, -3, 0, 0, 0], [0, 0, 0, 0, 5, -2], [0, -5, 0, 0, -1, -1], [0, -4, 0, 0, 1, 0], [2, 0, -2, 2, 0, 0], [-3, 0, 5, -1, 0, 0], [0, -4, 0, 0, -4, -3], [-1, 0, -2, 2, 0, 0], [0, 1, 0, 0, 4, 2], [1, 0, 0, -5, 0, 0], [0, -1, 0, 0, 10, 0], [-3, 0, -3, -2, 0, 0], [0, -1, 0, 0, -1, 5], [1, 0, -4, 5, 0, 0], [0, 8, 0, 0, -3, -3], [0, 3, 0, 0, -9, -2], [-4, 0, 5, 2, 0, 0], [0, -3, 0, 0, 6, 3], [0, 7, 0, 0, -3, -2], [6, 0, 3, 4, 0, 0], [0, 1, 0, 0, -10, 2], [9, 0, -2, -6, 0, 0], [-8, 0, 5, -4, 0, 0], [-10, 0, -3, 3, 0, 0], [0, 3, 0, 0, 7, -4], [0, -1, 0, 0, 3, -3], [0, -3, 0, 0, 0, 2], [0, 2, 0, 0, -11, 5], [5, 0, -8, -2, 0, 0], [-7, 0, 0, 0, 0, 0], [-2, 0, -4, 1, 0, 0], [0, 1, 0, 0, -2, -2], [0, -6, 0, 0, -6, 1], [2, 0, 2, -9, 0, 0], [17, 0, 0, -4, 0, 0], [0, -6, 0, 0, -1, 8], [0, -1, 0, 0, 4, -4], [-8, 0, 3, 5, 0, 0], [0, 0, 0, 0, 6, -11], [13, 0, 5, 3, 0, 0], [-8, 0, 7, 0, 0, 0], [6, 0, 3, -9, 0, 0], [0, 12, 0, 0, -3, -1], [0, -9, 0, 0, 3, -1], [28, 0, -1, -1, 0, 0], [0, 0, 2, 5, 0, 0], [0, -4, 0, 0, 5, -4], [4, 0, 8, -2, 0, 0], [0, -5, 0, 0, -2, -8], [0, 11, 0, 0, -17, -2], [0, -7, 0, 0, -1, -10], [19, 0, -1, -1, 0, 0], [0, -6, 0, 0, 8, -3], [0, 1, 0, 0, 14, -10], [0, 0, -1, -11, 0, 0], [0, 5, 0, 0, -1, 3], [0, 6, 0, 0, -6, -9], [0, 0, -5, 5, 0, 0], [0, 11, 0, 0, 4, -1], [15, 0, 3, -7, 0, 0], [0, -13, 0, 0, -4, -8], [0, -5, 0, 0, -17, 1], [-7, 0, 17, -3, 0, 0], [0, 3, 0, 0, 15, -1], [13, 0, 2, -16, 0, 0], [0, 1, 0, 0, 6, -1], [9, 0, 0, 3, 0, 0], [-1, 0, -1, -4, 0, 0], [-2, 0, -1, 11, 0, 0], [7, 0, 1, 7, 0, 0], [-8, 0, 8, 1, 0, 0], [0, 1, 0, 0, -17, -4], [-10, 0, -4, 13, 0, 0], [0, -3, 0, 0, 18, -6], [-3, 0, 9, 7, 0, 0], [0, -1, 0, 0, 23, -1], [8, 0, 13, -1, 0, 0], [0, 9, 0, 0, -10, -1], [0, 9, 0, 0, -12, 2], [12, 0, -6, 6, 0, 0], [0, 5, 0, 0, -22, 8], [4, 0, -1, -8, 0, 0], [13, 0, 6, -7, 0, 0], [0, 10, 0, 0, -13, -7], [8, 0, -11, 12, 0, 0], [-24, 0, -6, 6, 0, 0], [0, -4, 0, 0, -3, 11], [-8, 0, 4, 13, 0, 0], [0, 7, 0, 0, 6, 5], [0, 10, 0, 0, -25, -7], [0, 4, 0, 0, 2, 1], [4, 0, -10, -14, 0, 0], [-12, 0, 11, 2, 0, 0], [0, -6, 0, 0, -11, 9], [0, 0, 0, 0, -36, 3], [0, -11, 0, 0, -7, 11], [2, 0, -8, 1, 0, 0], [-22, 0, 3, -2, 0, 0], [0, 7, 0, 0, 30, 0], [0, -5, 0, 0, 16, -11], [0, 14, 0, 0, -8, -10], [6, 0, -11, -2, 0, 0], [28, 0, -5, 2, 0, 0], [13, 0, -4, 0, 0, 0], [0, 0, 0, 0, -22, 0], [0, 5, 0, 0, -5, 10], [0, -1, 0, 0, -11, 6], [24, 0, -15, 3, 0, 0], [0, 0, -1, -1, 0, 0], [0, -7, 0, 0, 9, 12], [0, 0, 0, 0, -12, 9], [0, 10, 0, 0, 11, -9], [6, 0, -4, -16, 0, 0], [14, 0, 7, -7, 0, 0], [25, 0, 3, -15, 0, 0], [-13, 0, 1, 11, 0, 0], [0, 4, 0, 0, 21, 5], [0, -10, 0, 0, -9, 2], [-8, 0, 21, -9, 0, 0], [0, -11, 0, 0, 11, 8], [-13, 0, -3, -10, 0, 0], [0, -5, 0, 0, 3, -2], [-10, 0, 13, -14, 0, 0], [-6, 0, -14, 0, 0, 0], [0, 1, 0, 0, -7, -4], [0, 1, 0, 0, 21, -5], [0, 11, 0, 0, 16, 6], [-5, 0, 11, -9, 0, 0], [10, 0, -23, 7, 0, 0], [4, 0, 5, 6, 0, 0], [0, 3, 0, 0, 8, -3], [0, -17, 0, 0, -16, -7], [-10, 0, 2, 4, 0, 0], [3, 0, -21, 7, 0, 0], [0, -14, 0, 0, 16, -3], [0, 5, 0, 0, -26, 5], [-12, 0, 19, 8, 0, 0], [0, -6, 0, 0, 39, 2], [0, 16, 0, 0, -12, 4], [23, 0, -5, -6, 0, 0], [0, 14, 0, 0, 9, -7], [0, -10, 0, 0, 1, 17], [0, -3, 0, 0, 19, 6], [-11, 0, -7, -11, 0, 0], [10, 0, 15, 7, 0, 0], [0, 0, -9, 18, 0, 0], [0, 18, 0, 0, -29, -9], [26, 0, -11, 2, 0, 0], [0, 17, 0, 0, 1, -14], [0, -9, 0, 0, 18, 0], [0, 1, 0, 0, -30, 3], [0, 0, -6, -8, 0, 0], [0, -11, 0, 0, -1, 7], [0, -4, 0, 0, 25, 4], [-6, 0, -13, -16, 0, 0], [0, -13, 0, 0, -13, -4], [-20, 0, 12, 9, 0, 0], [0, 12, 0, 0, 18, -4], [-1, 0, 19, -7, 0, 0], [5, 0, 18, -20, 0, 0], [14, 0, -23, 7, 0, 0], [0, 0, 0, 0, 14, -6], [-19, 0, -19, -12, 0, 0], [11, 0, -17, -1, 0, 0], [-6, 0, 20, -8, 0, 0], [12, 0, 9, 9, 0, 0], [0, 10, 0, 0, -20, -1], [-12, 0, -8, 20, 0, 0], [0, 8, 0, 0, -13, -10], [0, 0, -27, -6, 0, 0], [0, -7, 0, 0, -3, 13], [0, -7, 0, 0, -29, -1], [0, -12, 0, 0, 15, 1], [33, 0, -12, 3, 0, 0], [0, 8, 0, 0, 26, -13], [0, -10, 0, 0, 24, 9], [-19, 0, -4, -7, 0, 0], [2, 0, -8, 17, 0, 0], [0, 0, 0, 0, -19, 12], [0, -12, 0, 0, 36, 1], [-14, 0, -17, 16, 0, 0], [33, 0, 0, -8, 0, 0], [0, 4, 0, 0, 23, 8], [0, -10, 0, 0, -26, 6], [21, 0, -16, 14, 0, 0], [0, -7, 0, 0, -3, 0], [0, 10, 0, 0, 8, -13], [0, 2, 0, 0, 4, 0], [33, 0, 9, 7, 0, 0], [-30, 0, 11, 5, 0, 0], [0, -6, 0, 0, -23, 10], [-26, 0, 9, -5, 0, 0], [-14, 0, 29, -15, 0, 0], [0, -14, 0, 0, -16, 21], [11, 0, 5, 17, 0, 0], [0, 11, 0, 0, -14, -13], [-11, 0, 26, 3, 0, 0], [3, 0, 11, 12, 0, 0], [0, -17, 0, 0, 12, 9], [-13, 0, -14, 2, 0, 0], [0, -5, 0, 0, 13, 18], [0, 6, 0, 0, 39, 1], [-14, 0, 17, -2, 0, 0], [-30, 0, -7, 9, 0, 0], [0, 9, 0, 0, 34, -12], [21, 0, -24, 1, 0, 0], [0, -6, 0, 0, 18, -5], [0, 14, 0, 0, -17, -13], [-2, 0, -14, 13, 0, 0], [28, 0, 3, 1, 0, 0], [27, 0, -15, 11, 0, 0], [0, 33, 0, 0, -28, -9], [0, -7, 0, 0, -11, -8], [35, 0, -4, 0, 0, 0], [0, -14, 0, 0, -19, 18], [-9, 0, -9, -6, 0, 0], [0, -16, 0, 0, -22, -4], [-26, 0, -1, -1, 0, 0], [0, -17, 0, 0, -29, 7], [-19, 0, -1, -10, 0, 0], [32, 0, 9, -12, 0, 0], [-11, 0, -1, 26, 0, 0], [0, -10, 0, 0, -21, 8], [0, 11, 0, 0, 10, 7], [-9, 0, 22, 6, 0, 0], [0, -18, 0, 0, -6, 13], [4, 0, -13, 1, 0, 0], [42, 0, -3, -10, 0, 0], [0, 9, 0, 0, 18, -1], [1, 0, -8, -2, 0, 0], [0, -10, 0, 0, 6, -16], [-19, 0, -7, -9, 0, 0], [0, -21, 0, 0, 6, -3], [-15, 0, 15, -3, 0, 0], [0, 14, 0, 0, 20, 8], [-21, 0, 3, -1, 0, 0], [46, 0, -6, -5, 0, 0], [0, -14, 0, 0, 2, 14], [0, -7, 0, 0, 23, -1], [21, 0, -12, -16, 0, 0], [0, -7, 0, 0, -20, -14], [-16, 0, 20, 14, 0, 0], [0, -25, 0, 0, -5, 1], [13, 0, -19, 3, 0, 0], [37, 0, 1, -8, 0, 0], [0, -4, 0, 0, 11, 11], [0, -35, 0, 0, 23, 8], [0, 9, 0, 0, -17, 15], [0, 12, 0, 0, 17, -12], [0, 3, 0, 0, -9, 10], [36, 0, 16, 0, 0, 0], [0, 16, 0, 0, -20, -5], [0, 1, 0, 0, -61, 5], [-19, 0, -7, 11, 0, 0], [-6, 0, -3, 15, 0, 0], [-11, 0, 5, -1, 0, 0], [0, 7, 0, 0, -41, 13], [0, -16, 0, 0, -14, 15], [32, 0, 12, 3, 0, 0], [0, -8, 0, 0, 27, -8], [0, -13, 0, 0, 54, -1], [22, 0, -18, 0, 0, 0], [0, 4, 0, 0, -12, -3], [0, -23, 0, 0, 10, 1], [0, -4, 0, 0, 19, -12], [28, 0, 15, 8, 0, 0], [-13, 0, 3, -5, 0, 0], [45, 0, -18, 8, 0, 0], [0, 11, 0, 0, -20, 2], [30, 0, -4, -7, 0, 0], [0, -7, 0, 0, 13, 5], [-23, 0, -8, 3, 0, 0], [0, 0, 0, 0, 10, 6], [8, 0, -14, -2, 0, 0], [0, 6, 0, 0, 38, -25], [0, -21, 0, 0, 42, 10], [18, 0, 4, -1, 0, 0], [-14, 0, 2, 5, 0, 0], [10, 0, -4, -23, 0, 0], [0, 13, 0, 0, -10, -2], [0, -8, 0, 0, -7, 14], [-25, 0, 16, 12, 0, 0], [0, 12, 0, 0, 17, 1], [-27, 0, 27, -1, 0, 0], [-37, 0, 18, -12, 0, 0], [0, 4, 0, 0, 20, -30], [-19, 0, -17, 27, 0, 0], [0, 5, 0, 0, -4, 10], [0, 20, 0, 0, -39, -7], [0, 7, 0, 0, 44, -16], [-8, 0, 21, -7, 0, 0]]
