
# q-expansion of newform 1755.2.i.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 = 1755
weight = 2
poly_data = [268, 770, 1103, 1215, 1118, 859, 551, 326, 231, -19, 178, -107, 96, -44, 20, -5, 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], [[-6293, -3336, -4420, -3205, -2087, -1228, -677, -351, -120, -139, 39, -68, 28, -16, 4, -1], 2187], [[194118, 436375, 567013, 509676, 354893, 312111, 42206, 152463, -18231, 23394, 24185, -18634, 14280, -4880, 1221, -53], 68526], [[2021, 70, -607, -2345, -2093, -1580, -917, -447, -840, 757, -874, 469, -217, 50, -7, -1], 729], [[-568472, -2176341, -3007411, -3222844, -2132363, -2073673, -271472, -1222965, 363147, -417232, 20721, 9478, -42716, 11696, -2825, -547], 205578], [[492218, 860673, 651970, 555574, 364985, 163915, 470, 265752, -309462, 338866, -241101, 137264, -56011, 16531, -2977, 280], 102789], [[53568, 322685, 423533, 415092, 253195, 367611, -81968, 281373, -144885, 131148, -53597, 28186, -7602, 2996, -633, 197], 68526], [[299584, 797105, 1149565, 926282, 875795, 352535, 484382, -138783, 378201, -263068, 236395, -127504, 62410, -19076, 4147, -263], 68526], [[-3302, -4328, -6168, -4999, -3732, -2383, -1407, -780, -207, -403, 170, -219, 97, -51, 13, -3], 729], [[93260, 216123, 162604, 179317, 47699, 92848, -39292, 54630, -39966, 25879, -8913, -2101, 4688, -3278, 1103, -197], 34263], [[3587, 8634, 5971, -1982, 1643, -4508, -3007, 2448, -6243, 4663, -3357, 1253, -331, -53, 44, -14], 2187], [[610868, 1540755, 2034148, 1685476, 1392206, 552091, 721403, -211347, 561027, -433283, 410358, -248173, 127925, -44171, 10427, -1076], 102789], [[-3849, -6526, -10309, -9222, -7880, -5211, -3245, -1974, -165, -1425, 826, -833, 390, -187, 48, -10], 729], [[214288, 611973, 949991, 1097747, 951484, 658346, 543790, 105723, 252699, -22324, 47409, 30967, -21635, 18674, -5918, 1481], 102789], [[-2714, -12516, -31426, -40489, -35156, -28558, -18500, -8262, -6591, -2350, 426, -2744, 1591, -1030, 307, -70], 2187], [[-181150, -457891, -710436, -837740, -722337, -556832, -278193, -276987, 22059, -167819, 88954, -89361, 42641, -21726, 5900, -1260], 34263]]

hecke_ring_character_values = [[326, [-1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], [352, [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], [1081, [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]]
aps_data = [[0, -1, 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], [-1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 1, 1, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, -1, -1], [1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, -1, 0, -1, 0, 0, 0, 0, 0, -1, 2, 0, -1, 0, 0, -1], [-2, -1, 0, 1, 0, 0, 0, 0, 1, -1, 1, 0, 1, 0, 1, -1], [2, 0, 0, 0, -1, -2, 0, 0, 0, 0, 2, 0, 0, -1, 0, 0], [0, 2, 2, 1, -1, 0, -2, 0, 0, 0, 0, 0, 0, 1, 1, 0], [5, 0, -3, 0, -1, -2, -2, 0, 0, 2, 2, 1, 0, -1, 0, 0], [1, 2, 0, -1, 0, 0, 0, 0, 1, 0, -1, 0, -1, 0, 0, 0], [-1, 0, 2, 0, 1, -1, 1, 1, 0, 0, 1, -1, 0, -2, 0, 0], [0, 0, -2, -3, 3, 2, 0, 1, -1, 0, 0, -1, 1, 0, 0, 0], [0, -2, -1, -2, 2, -2, 2, 2, -2, 0, 0, -2, 2, 0, 0, 2], [-2, -2, 0, 2, 0, 0, 0, 0, -1, -2, 2, 0, 2, 0, 1, -2], [2, 0, -3, 0, -1, 1, -1, 1, 0, -2, -1, -1, 0, 0, 0, 0], [0, 0, 2, 2, -2, -4, 0, 3, 1, 0, 0, 1, 3, 1, 1, 0], [3, 0, -5, 0, -3, 2, 1, 2, 0, -1, -2, -2, 0, 1, 0, 0], [-2, -2, 0, 0, 0, 0, 0, 0, -2, 2, -1, 0, -1, 0, 1, 2], [1, 2, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0, 2, 0, -3, 0], [0, -6, -2, -1, 1, -1, 6, 1, -1, 0, 0, -1, 1, 0, 0, -2], [0, -1, -1, -2, 2, 1, 1, -1, 0, 0, 0, 0, -1, -2, -2, 0], [0, -1, 0, 2, 0, 0, 0, 0, 3, -3, 1, 0, 1, 0, 1, -3], [0, 3, 6, 1, -1, 0, -3, 0, 2, 0, 0, 2, 0, -3, -3, 1], [0, 2, 5, 3, -3, -4, -2, 2, -1, 0, 0, -1, 2, 1, 1, -1], [2, 0, -4, 0, 0, 2, 0, 0, 0, -1, -2, 1, 0, -2, 0, 0], [3, 3, 0, 3, 0, 0, 0, 0, 3, 2, 0, 0, -2, 0, -1, 2], [0, 4, 0, -1, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0], [-1, 0, 1, 0, 3, 0, 0, 0, 0, -1, 0, -1, 0, -1, 0, 0], [-4, 0, 0, 0, 0, 0, 0, 0, -1, 1, -3, 0, -2, 0, -1, 1], [3, 0, 2, 0, 3, -5, 0, 1, 0, 1, 5, -2, 0, 2, 0, 0], [0, 2, -1, 2, -2, 1, -2, -3, 2, 0, 0, 2, -3, -2, -2, 0], [7, 0, -5, 0, 2, -2, -2, 1, 0, 0, 2, -2, 0, -1, 0, 0], [-3, 0, 0, 0, 1, 3, 2, -2, 0, 1, -3, 3, 0, -1, 0, 0], [0, 4, 10, -1, 1, -1, -4, 1, 1, 0, 0, 1, 1, -1, -1, 2], [5, 0, -1, 0, 0, -4, -2, 1, 0, -6, 4, -3, 0, -4, 0, 0], [-6, -5, 0, 1, 0, 0, 0, 0, 2, -3, 3, 0, 1, 0, 3, -3], [1, 0, -4, 0, -1, 3, 6, 2, 0, 1, -3, 0, 0, 3, 0, 0], [0, -1, -6, 2, -2, 3, 1, 1, -2, 0, 0, -2, 1, 1, 1, 3], [-2, -5, 0, -2, 0, 0, 0, 0, 2, 1, 5, 0, -2, 0, 1, 1], [-5, 3, 0, 1, 0, 0, 0, 0, 4, -1, -1, 0, -3, 0, -3, -1], [0, -2, 1, -1, 1, 5, 2, -4, 3, 0, 0, 3, -4, 4, 4, -1], [2, 0, -3, 0, -3, 1, 3, 1, 0, -2, -1, -2, 0, 4, 0, 0], [4, 1, 0, -1, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 1], [-5, -5, 0, -4, 0, 0, 0, 0, -3, -4, 3, 0, 1, 0, 3, -4], [6, 0, -6, 0, 2, 0, 4, 4, 0, 3, 0, -2, 0, 4, 0, 0], [0, 5, 7, 7, -7, -3, -5, 0, 0, 0, 0, 0, 0, -3, -3, -1], [0, 4, -3, 4, -4, 3, -4, 1, 0, 0, 0, 0, 1, 1, 1, 4], [-1, 0, -1, 0, 0, 2, 4, 0, 0, 1, -2, 0, 0, 3, 0, 0], [-2, 6, 0, 0, 0, 0, 0, 0, 0, -3, -1, 0, -1, 0, -1, -3], [1, 0, 1, 0, -3, -2, -5, 1, 0, -4, 2, 2, 0, 2, 0, 0], [0, -9, 2, -6, 6, 2, 9, -2, -5, 0, 0, -5, -2, -1, -1, 5], [0, 10, 0, 3, 0, 0, 0, 0, 1, 0, -4, 0, 1, 0, -7, 0], [0, 0, 0, 0, 0, 0, 2, -2, 0, -1, 0, 2, 0, 0, 0, 0], [0, -1, 10, 0, 0, 2, 1, -5, 2, 0, 0, 2, -5, 2, 2, 5], [7, 3, 0, -5, 0, 0, 0, 0, -3, -1, -5, 0, 1, 0, -4, -1], [-3, -2, 0, -1, 0, 0, 0, 0, 0, 5, 1, 0, -1, 0, 3, 5], [0, -8, -5, -4, 4, 5, 8, -3, 0, 0, 0, 0, -3, -1, -1, 5], [0, 1, 3, 0, 0, -4, -1, 3, 3, 0, 0, 3, 3, -1, -1, -2], [2, 0, 2, 0, 1, -4, 4, 4, 0, 2, 4, -1, 0, -4, 0, 0], [-11, 0, 12, 0, -3, -1, -3, -6, 0, -3, 1, -1, 0, -1, 0, 0], [-2, 1, 0, 0, 0, 0, 0, 0, -3, 0, 2, 0, 0, 0, -3, 0], [2, 0, -8, 0, 3, 6, 0, -5, 0, -1, -6, 1, 0, -1, 0, 0], [0, 7, 2, 3, -3, 5, -7, -4, 4, 0, 0, 4, -4, -1, -1, -1], [0, -3, 12, -1, 1, 2, 3, -2, -3, 0, 0, -3, -2, -3, -3, 6], [0, -4, 0, 1, -1, 6, 4, 3, 6, 0, 0, 6, 3, 2, 2, -4], [-15, 0, 17, 0, -1, -2, -6, 0, 0, -1, 2, 1, 0, 2, 0, 0], [0, 0, 1, 0, -5, -1, 0, 4, 0, 2, 1, 0, 0, 2, 0, 0], [0, 1, 1, 0, 0, 4, -1, 1, 0, 0, 0, 0, 1, 1, 1, 1], [0, 3, 5, 4, -4, -3, -3, -4, -2, 0, 0, -2, -4, -8, -8, -1], [-13, 3, 0, 5, 0, 0, 0, 0, 5, -1, -3, 0, 4, 0, 4, -1], [0, -2, 11, -2, 2, 2, 2, 3, -3, 0, 0, -3, 3, 3, 3, 6], [-1, 0, 2, 0, 5, -1, 4, -1, 0, 1, 1, -2, 0, 4, 0, 0], [6, 2, 0, -7, 0, 0, 0, 0, -2, 3, 0, 0, 1, 0, -4, 3], [1, 0, 1, 0, -2, -2, -2, 1, 0, -9, 2, -5, 0, -4, 0, 0], [0, -5, -5, -5, 5, -4, 5, 4, -1, 0, 0, -1, 4, -1, -1, 0], [-12, 7, 0, -2, 0, 0, 0, 0, 2, -2, -3, 0, -2, 0, -1, -2], [-15, 0, 9, 0, 3, 6, 1, 1, 0, -7, -6, -2, 0, 2, 0, 0], [11, 0, -14, 0, -3, 3, 4, -3, 0, 2, -3, -1, 0, 1, 0, 0], [4, 0, -9, 0, 7, 5, 3, 1, 0, -1, -5, 2, 0, -1, 0, 0], [0, 6, 17, -1, 1, -3, -6, 2, 0, 0, 0, 0, 2, 2, 2, 8], [-6, 11, 0, 5, 0, 0, 0, 0, 0, -4, -2, 0, -2, 0, 2, -4], [1, -3, 0, 0, 0, 0, 0, 0, 1, -10, 0, 0, 2, 0, 3, -10], [0, 4, 9, -3, 3, -2, -4, 4, -2, 0, 0, -2, 4, 3, 3, -2], [0, -1, -9, 3, -3, -3, 1, 0, -2, 0, 0, -2, 0, -4, -4, -1], [-15, 5, 0, 1, 0, 0, 0, 0, 3, -8, -7, 0, -2, 0, -1, -8], [0, 1, 8, -2, 2, -3, -1, -1, 2, 0, 0, 2, -1, -3, -3, 4], [0, 1, 3, -5, 5, -4, -1, 2, 4, 0, 0, 4, 2, 1, 1, -1], [2, 0, -1, 0, 6, -1, 2, -5, 0, 1, 1, 2, 0, 0, 0, 0], [8, 10, 0, 0, 0, 0, 0, 0, 2, 5, -1, 0, -3, 0, 1, 5], [0, 4, -2, -1, 1, 6, -4, -5, -3, 0, 0, -3, -5, 3, 3, 0], [10, 2, 0, -13, 0, 0, 0, 0, -4, 7, 1, 0, -2, 0, 2, 7], [3, 0, -4, 0, -3, 1, -2, -1, 0, -1, -1, -2, 0, -1, 0, 0], [-3, 0, -5, 0, 0, 8, 9, 0, 0, 4, -8, 3, 0, 2, 0, 0], [9, -8, 0, -3, 0, 0, 0, 0, -3, 4, -3, 0, 2, 0, 5, 4], [1, 0, 3, 0, -5, -4, -4, -3, 0, 5, 4, 4, 0, 0, 0, 0], [-7, -8, 0, -1, 0, 0, 0, 0, 1, -4, 5, 0, 4, 0, 7, -4], [9, 1, 0, 4, 0, 0, 0, 0, -3, 7, 1, 0, 2, 0, 2, 7], [-10, -4, 0, 5, 0, 0, 0, 0, 0, -3, 4, 0, -2, 0, 3, -3], [0, -4, -3, -2, 2, 3, 4, -2, 1, 0, 0, 1, -2, 5, 5, 2], [13, 7, 0, 2, 0, 0, 0, 0, 2, 6, 4, 0, 0, 0, -4, 6], [11, 0, -16, 0, -4, 5, 1, -2, 0, 6, -5, 0, 0, 6, 0, 0], [0, -6, -4, 0, 0, 8, 6, -4, 4, 0, 0, 4, -4, 7, 7, 1], [0, 0, -4, 0, 7, 4, -3, -8, 0, -1, -4, 5, 0, 0, 0, 0], [5, 10, 0, 6, 0, 0, 0, 0, 6, 1, 5, 0, -4, 0, -1, 1], [0, -7, -10, 1, -1, -4, 7, 5, -1, 0, 0, -1, 5, 5, 5, -1], [9, 3, 0, -2, 0, 0, 0, 0, -4, -3, 5, 0, -1, 0, 1, -3], [-4, 0, 1, 0, 7, 3, -4, 4, 0, 0, -3, -5, 0, 2, 0, 0], [0, -2, -8, 7, -7, 1, 2, -3, 6, 0, 0, 6, -3, 3, 3, 5], [5, 0, -2, 0, 10, -3, 4, 7, 0, -2, 3, -6, 0, -5, 0, 0], [-8, -7, 0, 4, 0, 0, 0, 0, -3, -2, 3, 0, 0, 0, -1, -2], [-1, 0, -1, 0, 5, 2, 2, -6, 0, 2, -2, 1, 0, 4, 0, 0], [0, -1, 17, -3, 3, -2, 1, 5, -4, 0, 0, -4, 5, -12, -12, 0], [-6, 7, 0, 3, 0, 0, 0, 0, 0, 9, 4, 0, 2, 0, -2, 9], [0, -4, 4, -1, 1, 3, 4, -4, -2, 0, 0, -2, -4, -7, -7, -1], [4, 0, 1, 0, 4, -5, 4, 4, 0, -2, 5, -6, 0, 7, 0, 0], [13, 5, 0, 0, 0, 0, 0, 0, -2, 1, -4, 0, -1, 0, -1, 1], [-5, 0, 9, 0, 0, -4, -3, 0, 0, 3, 4, 4, 0, -1, 0, 0], [0, 2, -3, -3, 3, -4, -2, 2, 6, 0, 0, 6, 2, 6, 6, -9], [2, 0, -1, 0, 4, -1, 5, 0, 0, -2, 1, 3, 0, -8, 0, 0], [0, -5, 0, -4, 4, -1, 5, -1, -2, 0, 0, -2, -1, 5, 5, -3], [0, -15, -10, -2, 2, 6, 15, 1, 2, 0, 0, 2, 1, 6, 6, -7], [1, -9, 0, -4, 0, 0, 0, 0, 1, 3, 15, 0, -3, 0, -1, 3], [0, -3, 14, 0, 0, -3, 3, 1, -7, 0, 0, -7, 1, -9, -9, -5], [-10, -2, 0, -5, 0, 0, 0, 0, -3, -5, 1, 0, 7, 0, 0, -5], [0, 6, 7, 13, -13, -8, -6, -1, 1, 0, 0, 1, -1, 5, 5, -3], [-3, -1, 0, 3, 0, 0, 0, 0, 7, 8, -4, 0, 1, 0, -5, 8], [0, -2, 20, 5, -5, -7, 2, 4, -4, 0, 0, -4, 4, 2, 2, -3], [10, 0, -2, 0, -5, -8, -2, 3, 0, -1, 8, 2, 0, -5, 0, 0], [4, -8, 0, -10, 0, 0, 0, 0, 5, 2, 4, 0, 0, 0, 1, 2], [5, 0, -8, 0, 2, 3, 5, 0, 0, -9, -3, 8, 0, -3, 0, 0], [9, 0, -7, 0, 1, -2, 8, 6, 0, -1, 2, 0, 0, 9, 0, 0], [19, 4, 0, 3, 0, 0, 0, 0, 1, 10, 3, 0, 1, 0, -4, 10], [6, 0, -6, 0, -1, 0, 0, -8, 0, -2, 0, -5, 0, -8, 0, 0], [4, 0, -4, 0, 5, 0, 4, 0, 0, -6, 0, -2, 0, -1, 0, 0], [-8, 10, 0, 9, 0, 0, 0, 0, 6, -7, -3, 0, 4, 0, -3, -7], [9, 0, -18, 0, 0, 9, 3, -5, 0, -4, -9, 6, 0, 3, 0, 0], [1, 0, 11, 0, 1, -12, -3, 7, 0, 4, 12, -4, 0, 0, 0, 0], [-8, -2, 0, 7, 0, 0, 0, 0, -6, 1, 2, 0, -5, 0, 4, 1], [1, -4, 0, -2, 0, 0, 0, 0, 0, 8, 1, 0, -10, 0, -10, 8], [0, -2, -3, -8, 8, 13, 2, -2, -4, 0, 0, -4, -2, 5, 5, -1], [5, 0, 1, 0, 4, -6, 7, 3, 0, -4, 6, -5, 0, 5, 0, 0], [8, 2, 0, 11, 0, 0, 0, 0, 8, -6, 3, 0, -3, 0, -3, -6], [-5, -8, 0, -11, 0, 0, 0, 0, -5, 7, 5, 0, 4, 0, -2, 7], [0, -2, 4, -6, 6, -5, 2, 10, -6, 0, 0, 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