
# q-expansion of newform 2100.2.bi.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 = 2100
weight = 2
poly_data = [6561, -6561, 6561, -4374, 2592, -972, 0, 153, -167, 51, 0, -36, 32, -18, 9, -3, 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, 0, 0, 0, 0], 1], [[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 1], [[6561, -6561, 4374, -2592, 972, 0, -153, 167, -51, 0, 36, -32, 18, -9, 3, -1], 2187], [[20670066, -24411051, 19216764, -9118332, 3207744, -263352, -1296236, 938494, -339726, -105324, 151832, -121504, 76824, -29220, 6454, 1153], 609687], [[-54877662, 58844394, -39695913, 15893982, -4226967, -681303, 3585486, -1965202, 437691, 407982, -359337, 258388, -149400, 45069, -4206, -5395], 1829061], [[52235766, -81111456, 70727904, -40658976, 17467020, -4108284, -2590688, 3390626, -1670856, 72792, 430760, -438368, 304524, -149151, 46258, -5968], 1829061], [[-108434376, 151849728, -124258779, 69652251, -28159650, 5733957, 5956964, -6035418, 2652570, 73005, -848087, 786063, -534429, 241188, -71512, 6954], 1829061], [[185919057, -202887261, 135325242, -52760484, 13815873, 4191318, -12576744, 7041979, -1318053, -1588383, 1306899, -887410, 498699, -143820, 951, 23263], 5487183], [[276613947, -293889789, 211208067, -85116744, 26531901, 3420909, -17596443, 10069919, -2572404, -2035269, 1838910, -1365137, 767331, -242181, 32013, 24626], 5487183], [[47911338, -82166319, 76141134, -45346419, 19818666, -5344728, -2213422, 3602156, -1928373, 217959, 431323, -457733, 329391, -170922, 56618, -9427], 1829061], [[-33668865, -13079718, 41955408, -44623872, 25189920, -11666124, 4108284, 1677584, -2393970, 1366488, -72792, -215912, 247392, -197100, 95439, -28354], 5487183], [[60732990, -100513305, 87490611, -52309665, 23354802, -6075699, -2877419, 4183034, -2127084, 241278, 491762, -532664, 389808, -194877, 65950, -11554], 1829061], [[220725162, -195970509, 107884710, -28293624, -166968, 10498437, -15338910, 5622202, 221379, -2239380, 1323084, -769078, 367722, -13698, -50757, 35773], 5487183], [[33668865, 29541267, -55368522, 56817612, -32506164, 12817755, -4213662, -2550716, 2777847, -1637460, 42684, 321290, -315135, 242262, -110493, 35881], 5487183], [[-49687182, 13861206, 12957975, -25329105, 15176142, -8119533, 4215935, 476724, -1456956, 1097229, -176321, -49284, 108063, -114183, 62870, -21897], 1829061], [[-23491053, 8130132, 2059155, -7566768, 5458719, -3083533, 1822111, 101459, -481167, 424168, -93712, -302, 29256, -40300, 21901, -8452], 609687]]

hecke_ring_character_values = [[1051, [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], [701, [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], [1177, [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], [1501, [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0]]]
aps_data = [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, -1, 1], [-1, 1, -1, -1, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0], [0, 1, 0, -1, 1, 0, -1, -1, 1, 0, 0, 0, 0, 0, -1, -1], [-2, 1, 0, 0, 1, 1, -1, -2, 0, -1, 1, 0, 0, 0, 0, 0], [0, -1, 0, 0, 1, -1, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0], [0, 2, -1, 1, -1, 0, -1, -1, 0, -1, 0, 0, -2, -2, 0, 0], [0, -1, -2, -2, 1, -1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, -6, -6, 2, 1, -2, 0, 0, 2, 1, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, 2, 0, -1, 1, -1, 0, 1, 0, 0, 0, 0], [0, 1, 0, 0, -1, 1, -1, -1, 0, -1, 0, 0, 0, 0, 0, 0], [0, -2, 1, 0, 1, -1, 2, 1, -2, 2, 0, 1, 4, 2, -1, 1], [0, -2, 0, 1, 0, 1, 1, 0, -1, 1, 0, 0, 2, 0, -1, 1], [0, 1, 0, 1, -3, 2, 0, -1, -1, -1, 0, 0, -2, -4, 1, 1], [2, 2, 0, 0, -1, 2, -2, -1, 0, -2, -1, 0, 0, 0, 0, 0], [3, 3, -1, -1, -1, 3, 2, 1, 0, 1, -3, 0, 0, 0, 0, 0], [0, 0, -1, -3, 3, -2, -1, 0, 2, 2, 0, -2, 2, 2, -2, -2], [3, 1, -3, -3, -1, 1, 0, -1, 0, -1, 3, 0, 0, 0, 0, 0], [0, -2, -2, -2, 3, -2, -1, 2, 0, 1, 1, 0, 0, 0, 0, 0], [0, 3, 3, -1, 0, -1, 0, -1, 1, -1, 0, 1, 0, 0, 0, 0], [0, 0, -1, 0, 1, -1, 2, 1, 2, 2, 0, -1, 4, 2, 1, -1], [-1, 1, 0, 0, -1, 1, 1, -2, 0, 2, 2, 0, 0, 0, 0, 0], [0, -2, 1, -2, 2, -1, 1, 2, -1, 1, 0, 2, 2, 4, 1, -1], [-6, 0, 0, 0, 2, 0, 1, -1, 0, 1, 3, 0, 0, 0, 0, 0], [0, 1, 3, -2, 3, -2, -1, 0, 2, -1, 0, 1, 0, 4, -1, -1], [-5, 0, 1, 1, -2, 0, 2, 2, 0, 0, 5, 0, 0, 0, 0, 0], [0, 0, 0, -4, 2, 0, -2, -1, 2, 1, 0, -2, 0, 0, -2, -2], [5, -1, -1, -1, 0, -1, 0, 3, 0, 3, 0, 0, 0, 0, 0, 0], [0, -1, 0, -2, 1, -1, -1, 0, -2, 1, 0, 0, 0, 0, -2, 0], [0, 0, 2, 0, 0, -2, 1, 0, -2, 1, 0, 0, 2, 0, -2, 2], [-1, -5, 7, 7, -2, -5, 2, 2, 0, -2, 2, 0, 0, 0, 0, 0], [0, -2, 3, -4, 1, -5, 1, 4, -2, 1, 0, -1, 0, 4, 1, 1], [-1, 3, -2, -2, 1, 3, -3, -1, 0, -2, 1, 0, 0, 0, 0, 0], [-3, 1, -4, -4, -1, 1, -4, -1, 0, 3, -3, 0, 0, 0, 0, 0], [0, -5, 0, 0, 5, -5, 0, 5, 0, 0, -1, 0, 0, 0, 0, 0], [0, -4, -6, 4, 0, 6, 0, 0, 1, 0, 0, 1, 0, 0, 2, -2], [0, -3, 0, -2, 3, -3, 3, 2, -2, 5, 0, 0, 8, 4, -2, 0], [0, 4, 0, -2, 0, -2, 0, 0, 2, 0, 0, 0, 0, 0, 2, -2], [-6, 0, -1, -1, 1, 0, -1, 3, 0, -3, 12, 0, 0, 0, 0, 0], [0, 3, 3, -2, -2, 0, 0, -2, 0, 0, 0, -1, 0, -4, -1, 1], [6, -6, 4, 4, -2, -6, 1, 2, 0, -1, -6, 0, 0, 0, 0, 0], [0, -1, -1, 1, 2, -3, 2, 2, 2, 2, 0, -2, 4, 4, 0, 0], [3, 2, 4, 4, -2, 2, 1, -2, 0, -3, 3, 0, 0, 0, 0, 0], [1, 2, 1, 1, -1, 2, -1, -5, 0, -5, 0, 0, 0, 0, 0, 0], [-3, 5, -2, -2, -3, 5, 3, 2, 0, -2, 6, 0, 0, 0, 0, 0], [0, -1, 2, -2, -1, 1, 1, 1, -2, 0, 0, 2, 0, 0, 2, 0], [-2, -2, 0, 0, -2, -2, 3, 5, 0, 3, 1, 0, 0, 0, 0, 0], [0, -5, -5, 6, 2, 0, 0, 2, 0, 0, 0, -1, 0, 4, -1, 1], [0, 2, -3, 3, 1, -4, 1, 1, -2, 1, 0, 2, 2, 2, 0, 0], [-3, 1, 3, 3, -1, 1, 3, -1, 0, -4, -3, 0, 0, 0, 0, 0], [0, -3, -4, 2, 1, 3, -1, 1, 1, -1, 0, 1, -2, 2, 2, -2], [0, 4, -1, -2, 0, -3, -3, -1, 0, -1, 0, -1, -4, -2, -1, -1], [0, 2, -1, -1, 0, 0, -2, 0, 2, -2, 0, 0, -4, 0, 2, -2], [0, 2, 4, -5, 2, -1, -2, -2, 3, 0, 0, -2, 0, 0, -3, -1], [4, -6, 0, 0, -1, -6, -3, -2, 0, -3, -2, 0, 0, 0, 0, 0], [9, 1, -2, -2, 3, 1, 0, -3, 0, 3, -9, 0, 0, 0, 0, 0], [0, 3, -2, 6, -3, -1, 1, 0, -1, -2, 0, 1, -2, -2, 2, 2], [1, 3, 8, 8, -3, 3, 0, -3, 0, -3, 1, 0, 0, 0, 0, 0], [0, 3, -1, -3, -2, 1, 2, -1, 0, 3, 0, 0, 4, -4, 2, -2], [-2, -9, 4, 4, 5, -9, -5, -1, 0, 1, 4, 0, 0, 0, 0, 0], [0, 0, -2, 4, 0, -2, 0, 0, 2, 0, 0, -4, 0, 0, -2, 2], [-12, -4, 0, 0, 0, -4, -1, -1, 0, -1, 6, 0, 0, 0, 0, 0], [0, 1, 2, -4, -2, -1, 1, 1, -2, 1, 0, 0, 0, -2, 2, 0], [0, 2, -4, -4, 0, 2, 2, 2, 0, -2, 7, 0, 0, 0, 0, 0], [-16, 6, 6, 6, 0, 6, 0, -3, 0, -3, 0, 0, 0, 0, 0, 0], [0, 5, 2, 1, -1, 0, 0, -1, 3, -3, 0, 2, -2, 0, 3, -1], [4, -7, 3, 3, 4, -7, -4, -3, 0, 3, -8, 0, 0, 0, 0, 0], [0, 2, 3, 0, -2, -1, -1, -2, 3, -1, 0, -6, -2, -4, -3, 3], [0, 7, 11, -8, 3, -4, -2, -3, 4, -2, 0, 1, 0, 2, -3, -1], [-12, -1, 6, 6, 1, -1, 3, 1, 0, -2, -12, 0, 0, 0, 0, 0], [0, -4, 6, 6, 1, -4, 2, 3, 0, -2, -1, 0, 0, 0, 0, 0], [1, 6, 7, 7, 1, 6, 1, -5, 0, -5, 0, 0, 0, 0, 0, 0], [0, -2, 2, 6, -2, 4, 5, 1, 2, -1, 0, 2, 4, 2, 4, 2], [0, 1, 3, -4, 1, -8, 1, 1, -5, 4, 0, -2, 4, 0, -5, 3], [-8, 2, 0, 0, -5, 2, -1, 4, 0, -1, 4, 0, 0, 0, 0, 0], [0, 1, -1, -4, -2, 2, 1, 1, -2, 1, 0, 3, 0, -2, 5, -3], [7, 4, -12, -12, -4, 4, -2, -4, 0, -2, 7, 0, 0, 0, 0, 0], [0, -7, -7, 9, 0, 9, 0, -1, 1, -1, 0, 1, 0, 0, 0, 0], [-9, -1, 4, 4, 5, -1, 5, 2, 0, 2, 0, 0, 0, 0, 0, 0], [0, 5, 0, -6, 1, -7, 2, 1, 0, 5, 0, -2, 6, 0, 0, -2], [-6, 4, -5, -5, 1, 4, -1, 1, 0, -1, 12, 0, 0, 0, 0, 0], [18, -4, 0, 0, 5, -4, 3, -2, 0, 3, -9, 0, 0, 0, 0, 0], [0, -5, -8, 2, 1, 3, 1, 4, -2, 1, 0, 4, 0, 4, 6, -4], [0, 5, -5, 5, 0, -5, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0], [0, -1, 0, 0, 1, -1, -4, -3, 0, 4, -7, 0, 0, 0, 0, 0], [0, -4, -6, -2, 0, 0, 0, 3, -2, 3, 0, -2, 0, 0, 2, -2], [-13, 2, 1, 1, -1, 2, -1, -6, 0, -6, 0, 0, 0, 0, 0, 0], [0, 3, 2, 6, -3, 1, 3, 0, 2, -3, 0, 2, 0, 0, 4, 2], [0, -4, 3, -4, 4, -3, 2, 4, -1, 2, 0, 2, 4, 8, 1, -1], [14, -8, 3, 3, 2, -8, -3, -2, 0, -1, -14, 0, 0, 0, 0, 0], [0, -2, 2, 6, -4, 2, 4, 2, -2, -2, 0, 2, 0, 0, 4, 4], [0, 4, -8, -8, 0, 4, 2, 2, 0, -2, 7, 0, 0, 0, 0, 0], [0, 6, 7, -11, 1, -12, -1, 3, -2, 1, 0, -2, -2, 2, 0, 0], [0, -4, -4, -8, 6, -4, 0, 2, 0, 8, 0, -4, 8, 4, -4, -4], [0, 4, 4, -4, -4, 4, -2, -4, 0, -2, 0, 0, -4, -8, 0, 0], [4, 1, 0, 0, -4, 1, 1, 5, 0, 1, -2, 0, 0, 0, 0, 0], [0, -5, 4, 4, 3, -5, -2, 1, 0, 2, 7, 0, 0, 0, 0, 0], [2, 7, 7, 7, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 6, -2, -10, 2, -10, -5, 2, 0, 1, 0, -4, -6, 0, 0, -4], [0, 0, -8, 5, 0, 3, 0, 0, -3, 0, 0, 6, 0, 0, 3, -3], [0, 8, 7, -8, -1, 1, -1, -4, 2, -1, 0, 3, 0, -4, 1, -3], [11, 5, 0, 0, -5, 5, 4, 5, 0, -1, -11, 0, 0, 0, 0, 0], [-10, 3, 6, 6, -3, 3, 1, -3, 0, -4, -10, 0, 0, 0, 0, 0], [0, -5, 1, 3, 4, -3, -4, 3, 0, -5, 0, 0, -8, 8, -2, 2], [0, 2, -4, -4, 0, 2, -6, -2, 4, -2, 0, -4, -8, -4, 0, -4], [0, 9, 7, 6, -3, 2, 0, -3, 5, -9, 0, 6, -6, 0, 5, 1], [-3, -9, 11, 11, -2, -9, 2, -3, 0, 3, 6, 0, 0, 0, 0, 0], [-18, 1, 0, 0, 1, 1, -1, -2, 0, -1, 9, 0, 0, 0, 0, 0], [8, -12, 1, 1, 10, -12, -9, -10, 0, 1, -8, 0, 0, 0, 0, 0], [0, -4, 0, 4, 2, -4, 6, 5, -4, 3, 0, 4, 8, 8, 2, 2], [14, -9, -7, -7, 9, -9, 2, 9, 0, 7, 14, 0, 0, 0, 0, 0], [13, 0, -5, -5, -5, 0, -5, 1, 0, 1, 0, 0, 0, 0, 0, 0], [0, -1, -2, 4, -1, 7, -3, -1, 4, -6, 0, 2, -8, 0, 4, -2], [-5, 6, -14, -14, 8, 6, -8, -2, 0, 2, 10, 0, 0, 0, 0, 0], [0, -4, 2, 1, 0, -3, 3, 4, -1, 1, 0, -2, 2, 4, 1, 3], [0, 4, 1, 6, -1, 3, -2, -7, 4, -2, 0, -3, 0, -6, -7, 3], [0, 2, -5, 1, 5, -8, 1, 2, 5, 4, 0, -5, 6, 6, -2, -2], [-5, -1, -17, -17, 1, -1, -6, 1, 0, 7, -5, 0, 0, 0, 0, 0], [-5, 3, 0, 0, -3, 3, -3, -4, 0, -4, 0, 0, 0, 0, 0, 0], [0, -2, -2, 0, -1, 6, -2, -1, 4, -2, 0, -2, -4, -2, 2, -2], [0, 0, 0, 7, -2, 9, 5, -2, 5, -1, 0, 4, 6, 0, 5, -1], [-14, 3, 0, 0, 6, 3, 2, -4, 0, 2, 7, 0, 0, 0, 0, 0], [0, 3, -3, 7, -2, -3, 2, 1, -3, -1, 0, 3, 0, 0, 2, 2], [0, 0, -8, -8, 4, 0, -9, -5, 0, 9, -9, 0, 0, 0, 0, 0], [0, 1, 5, 8, -5, 4, 2, -1, -2, -6, 0, 5, -4, -2, 3, 5], [-5, -13, 3, 3, 10, -13, -10, -2, 0, 2, 10, 0, 0, 0, 0, 0], [22, -6, 0, 0, 3, -6, 2, -1, 0, 2, -11, 0, 0, 0, 0, 0], [-1, -10, 8, 8, -6, -10, 8, 6, 0, 2, 1, 0, 0, 0, 0, 0], [0, -3, 3, -3, 0, 3, 0, 0, -2, 0, 0, 2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 3, 3, 0, -3, -13, 0, 0, 0, 0, 0], [-15, 0, 0, 0, 0, 0, 0, 7, 0, 7, 0, 0, 0, 0, 0, 0], [0, 1, -1, -4, 4, -8, 2, 2, -2, 6, 0, -1, 8, 4, -3, -1], [6, -10, 18, 18, -8, -10, 8, 7, 0, -7, -12, 0, 0, 0, 0, 0], [0, -2, 0, 0, 2, -2, 1, 2, 0, 1, 0, 0, 2, 4, 0, 0], [-3, -7, -23, -23, 7, -7, -3, 7, 0, 10, -3, 0, 0, 0, 0, 0], [0, -3, -1, 9, -2, 7, 2, -4, 0, 0, 0, 0, 4, -4, -4, 4], [0, -4, -3, -5, 2, -4, -1, 2, -4, 5, 0, -4, 2, 0, -4, 0], [-15, 7, -13, -13, 6, 7, -6, -5, 0, 5, 30, 0, 0, 0, 0, 0], [0, 1, 7, 4, 3, -6, -2, -3, 4, -2, 0, -7, 0, 2, -11, 7], [19, 2, 1, 1, -4, 2, 7, 4, 0, 3, -19, 0, 0, 0, 0, 0], [0, 2, -2, -2, 2, -2, -2, -1, -8, 1, 0, 8, 0, 0, -2, -2], [1, 0, -2, -2, 0, 0, -1, 0, 0, 1, 1, 0, 0, 0, 0, 0], [0, -6, 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