
# q-expansion of newform 1155.2.c.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 = 1155
weight = 2
poly_data = [4, 0, 3428, 0, 22984, 0, 55281, 0, 63865, 0, 40640, 0, 15210, 0, 3426, 0, 456, 0, 33, 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, 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, 0, 0, 0, 0], 1], [[3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 1], [[-10112, 0, -3622164, 0, -17834990, 0, -30691092, 0, -24948019, 0, -10903803, 0, -2720792, 0, -388661, 0, -29599, 0, -932, 0], 171356], [[0, -52934572, 0, -263850094, 0, -580092520, 0, -801263181, 0, -625930321, 0, -271650511, 0, -67184034, 0, -9437765, 0, -701609, 0, -21445], 3255764], [[-12649508, -233638456, -207237480, -1064523572, -842531572, -1831293380, -1413013347, -1595603488, -1157693438, -813882325, -508016561, -262962518, -125789839, -55067041, -17627746, -7239623, -1304803, -539302, -39658, -17223], 6511528], [[1859432, 0, 46688632, 0, 163946632, 0, 328984885, 0, 341926714, 0, 186103148, 0, 55840854, 0, 9309882, 0, 807316, 0, 28357, 0], 3255764], [[-12324404, 242033984, -280996784, 1186256172, -1184482080, 2060015128, -2021456739, 1787344850, -1690185486, 890119595, -769151417, 271905052, -200454623, 52047565, -29875558, 6106853, -2366551, 402088, -77206, 11377], 6511528], [[-4197764, -176186716, -60866300, -1104999046, -114360874, -2550730974, -95870681, -2861591894, -38118635, -1782095748, -4471267, -655436372, 1509738, -145349694, 566385, -19061802, 68607, -1359828, 2923, -40638], 3255764], [[-4197764, 176186716, -60866300, 1104999046, -114360874, 2550730974, -95870681, 2861591894, -38118635, 1782095748, -4471267, 655436372, 1509738, 145349694, 566385, 19061802, 68607, 1359828, 2923, 40638], 3255764], [[-12324404, -242033984, -280996784, -1186256172, -1184482080, -2060015128, -2021456739, -1787344850, -1690185486, -890119595, -769151417, -271905052, -200454623, -52047565, -29875558, -6106853, -2366551, -402088, -77206, -11377], 6511528], [[0, 5043820, 0, 40268562, 0, 109059276, 0, 136502701, 0, 91834117, 0, 35730088, 0, 8272267, 0, 1123169, 0, 82492, 0, 2528], 171356], [[467236, 0, -14754680, 0, -71376026, 0, -119768558, 0, -95909007, 0, -41590554, 0, -10332043, 0, -1471065, 0, -111664, 0, -3503, 0], 171356], [[-8350020, 364697836, -324766228, 2490994876, -1153488952, 6285944028, -1622824553, 7744640527, -1136990015, 5254376982, -436700849, 2080024161, -95841406, 491154011, -11937193, 67999162, -781727, 5086207, -20801, 158482], 6511528], [[-8350020, -364697836, -324766228, -2490994876, -1153488952, -6285944028, -1622824553, -7744640527, -1136990015, -5254376982, -436700849, -2080024161, -95841406, -491154011, -11937193, -67999162, -781727, -5086207, -20801, -158482], 6511528], [[12865832, -769879164, 617774332, -5229989096, 2109983528, -12401839144, 2525053938, -13969193167, 1498448529, -8644761455, 504609204, -3152696619, 101264279, -694260760, 12008451, -90636213, 777136, -6452077, 21163, -192793], 6511528], [[0, -500823852, 0, -3430562652, 0, -7886314178, 0, -8634683824, 0, -5231616809, 0, -1878395562, 0, -408736717, 0, -52853791, 0, -3732736, 0, -110783], 3255764], [[0, 15121348, 0, 117183522, 0, 309342838, 0, 378817011, 0, 250554332, 0, 96286461, 0, 22096009, 0, 2980846, 0, 217877, 0, 6652], 171356], [[0, -14264568, 0, -117012166, 0, -309342838, 0, -378817011, 0, -250554332, 0, -96286461, 0, -22096009, 0, -2980846, 0, -217877, 0, -6652], 171356], [[0, -32364064, 0, -211022344, 0, -506157206, 0, -581983849, 0, -367228218, 0, -136061779, 0, -30327549, 0, -3995054, 0, -286263, 0, -8594], 171356]]

hecke_ring_character_values = [[386, [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], [232, [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], [661, [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], [211, [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]]
aps_data = [[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, 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, -1, 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, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [0, 1, 0, 0, -1, 0, 0, 1, 1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, -1], [-2, -1, 0, 0, 0, -1, 0, 0, -2, 1, 0, 0, 0, 0, 1, -1, 1, 0, -1, 1], [0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 1, 0, 0], [0, 1, 1, -1, 0, 1, -1, -1, 1, -2, -1, 0, 0, 1, 0, 1, -1, 0, 1, -1], [0, 0, 1, -2, 0, 0, 2, 1, -1, -1, 1, 0, 1, -1, -1, 0, 0, 0, 0, 0], [0, -1, 0, 0, -1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 1, -1, -2, 1], [4, 0, 1, 1, 0, 0, -1, 0, 0, 0, 0, 0, -2, 1, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, -1, -1], [0, 3, 0, 0, 1, 0, 0, 1, 1, -1, -1, 0, 0, 0, 0, 0, -1, 2, 1, 0], [0, 1, 0, 0, 1, 0, 0, -1, 2, -2, 1, -2, 0, 1, -1, 0, -2, 1, 1, 0], [-2, 0, -2, 1, 0, -2, 1, 0, -2, 2, 0, 0, 0, 0, 2, 0, 1, 0, -2, 1], [4, 0, 1, 0, 0, -2, 0, 1, -4, 0, 1, 0, 0, -1, 1, 0, 1, 0, -2, 1], [0, -1, 0, 0, 1, 0, 0, -1, -3, 3, 1, 0, 0, -1, 1, 0, 0, -1, -2, 1], [0, 0, -1, 1, 0, 0, 1, 0, 2, 2, 0, 0, -2, 1, 1, 0, 0, 0, 0, 0], [0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, -1, 0, 1], [-2, 2, 1, 2, 0, -2, 0, 1, -1, 1, 1, 0, -1, -1, 1, 2, 0, 0, -2, 0], [0, 0, 0, 0, -2, 0, 0, 0, 1, -1, 0, -2, 0, 1, -1, 0, 0, -3, 0, -2], [0, -2, 2, -2, 0, 0, -2, 0, -3, -1, 0, 0, 1, 0, 0, -2, 1, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, -2, 0, 0, 0, 0, 1, 4, 1, -1], [4, 0, 0, 0, 0, 0, -2, -2, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -4, 0, 0, 0, 0, 0, 1, -2, 2, -1, 4, 0, -2, 2, 0, 1, 0, -3, 1], [0, -4, 0, 0, -2, 0, 0, -1, 2, -2, 1, 4, 0, 1, -1, 0, -1, -1, 1, -1], [-4, 3, 2, -3, 0, 1, -1, 0, 4, -1, 0, 0, 0, 0, -1, 3, -2, 0, 1, -2], [0, -3, 0, 0, -1, 0, 0, 1, 1, -1, -1, 0, 0, 0, 0, 0, 0, -2, 0, 1], [0, 2, 0, 0, 0, 0, 0, -1, 1, -1, 1, -2, 0, 1, -1, 0, -3, 1, 0, 0], [4, 2, 1, 3, 0, -2, -1, 2, 0, 2, 2, 0, 0, -1, 1, 2, 0, 0, -2, 0], [0, -2, 0, 0, 0, 0, 0, -2, 1, -1, 2, 4, 0, 1, -1, 0, 0, -1, 3, -1], [0, -2, 0, 1, 0, -2, -1, 3, -3, 3, 3, 0, -3, 0, 2, -2, 2, 0, -2, 2], [-4, 1, 1, -3, 0, -1, 1, 3, 0, 1, 3, 0, -3, -1, 0, 1, 0, 0, -1, 0], [4, -1, 1, -3, 0, 1, -3, -1, 1, 0, -1, 0, -1, 1, 0, -1, 0, 0, 1, 0], [0, -2, 0, 0, 0, 0, 0, 1, -1, 1, -1, -4, 0, -1, 1, 0, 1, 1, 0, 0], [0, -4, 0, 0, -2, 0, 0, -3, -3, 3, 3, 0, 0, 0, 0, 0, 1, -4, -4, 2], [0, -1, 0, 0, 1, 0, 0, -1, -5, 5, 1, -2, 0, 0, 0, 0, 2, 2, 1, 0], [0, 2, 0, 0, 4, 0, 0, -4, -1, 1, 4, -4, 0, 1, -1, 0, 0, 1, 1, 1], [0, -2, -1, -3, 0, 2, 1, 0, 2, 0, 0, 0, 0, 1, -1, -2, 0, 0, 2, 0], [2, 2, -2, 2, 0, -2, -2, 2, 2, 4, 2, 0, 0, 0, 2, 2, 0, 0, -2, 0], [4, -2, 2, 0, 0, 2, 0, 0, -2, -4, 0, 0, 4, 0, -2, -2, 0, 0, 2, 0], [0, 0, 0, 0, -2, 0, 0, 0, 2, -2, 0, -4, 0, 2, -2, 0, 0, 2, 2, -2], [0, -1, 0, 0, 1, 0, 0, -1, -2, 2, 1, 2, 0, -1, 1, 0, 2, 1, 0, 3], [-6, -1, -1, -1, 0, 1, 1, 3, 1, 0, 3, 0, 3, 1, 0, -1, 0, 0, 1, 0], [8, 1, 1, -3, 0, 3, 1, -1, 3, -4, -1, 0, 3, -1, -4, 1, -2, 0, 3, -2], [0, 2, 0, 0, 2, 0, 0, -1, 0, 0, 1, 8, 0, 0, 0, 0, -1, 2, 3, 1], [0, -4, 0, 0, 4, 0, 0, -2, -4, 4, 2, 4, 0, -2, 2, 0, 0, 0, -1, 1], [-4, 3, -6, 1, 0, 1, 3, 0, 4, -1, 0, 0, 2, 0, -1, 3, -2, 0, 1, -2], [0, 0, 0, 0, 0, 0, 0, 1, 3, -3, -1, -4, 0, 1, -1, 0, -1, -1, -1, -1], [0, 1, -1, 0, 0, 1, -2, -2, 4, 1, -2, 0, 1, 1, 0, 1, -1, 0, 1, -1], [8, 0, 4, 1, 0, 0, 1, 2, 1, 1, 2, 0, -3, 2, 2, 0, 0, 0, 0, 0], [8, 1, 3, 1, 0, -1, -1, 4, -1, 0, 4, 0, -3, 1, 2, 1, 0, 0, -1, 0], [0, 1, 0, 0, -1, 0, 0, -1, 1, -1, 1, -2, 0, -2, 2, 0, 1, -2, -1, 0], [0, 1, 0, 0, 1, 0, 0, -1, -3, 3, 1, -2, 0, -1, 1, 0, 2, 5, -2, 1], [-6, 1, -5, -2, 0, -3, 0, 2, -1, 4, 2, 0, -1, -1, 2, 1, 1, 0, -3, 1], [4, -2, 1, 8, 0, -4, 2, 2, -6, 4, 2, 0, 0, -1, 3, -2, 3, 0, -4, 3], [0, -2, 0, 0, -2, 0, 0, 2, 1, -1, -2, -6, 0, 1, -1, 0, 0, -1, 1, 1], [-2, -2, 2, -3, 0, 2, -1, -5, -2, -4, -5, 0, 1, 2, 0, -2, 0, 0, 2, 0], [0, 3, 0, 0, -1, 0, 0, -1, 2, -2, 1, 2, 0, -1, 1, 0, 1, 1, 0, -1], [0, 3, 0, 0, -1, 0, 0, 1, -2, 2, -1, -6, 0, -1, 1, 0, 0, -3, -1, 0], [0, -4, 0, 0, 0, 0, 0, -2, -4, 4, 2, 0, 0, 0, 0, 0, 2, 0, -2, 2], [0, 1, 3, -3, 0, -5, -1, 1, -5, 4, 1, 0, -1, -1, 4, 1, 2, 0, -5, 2], [0, -4, 0, 0, 2, 0, 0, 3, -5, 5, -3, 8, 0, -3, 3, 0, 3, 1, -6, 2], [0, -4, 0, 0, 2, 0, 0, -2, 1, -1, 2, 4, 0, -1, 1, 0, 0, -3, -1, 1], [0, -2, 1, 5, 0, 0, 1, 4, -1, 1, 4, 0, -3, 1, 1, -2, 1, 0, 0, 1], [0, 0, 0, 0, 2, 0, 0, 3, -2, 2, -3, 0, 0, 0, 0, 0, -1, 2, -3, 1], [0, -3, 0, 0, 1, 0, 0, -1, 0, 0, 1, 6, 0, 1, -1, 0, -2, 1, 0, 1], [2, -1, 4, 4, 0, -1, -6, 0, 0, 3, 0, 0, -2, 0, 1, -1, 1, 0, -1, 1], [0, 4, 0, 0, 0, 0, 0, -1, -2, 2, 1, 4, 0, -2, 2, 0, 4, -2, 0, 0], [0, -2, -2, 2, 0, 4, 2, 0, -1, -7, 0, 0, -1, 0, -4, -2, -1, 0, 4, -1], [0, 2, 0, 0, 0, 0, 0, -3, 5, -5, 3, -10, 0, 1, -1, 0, -3, -5, 0, 0], [0, -2, 0, 0, -2, 0, 0, -1, 7, -7, 1, 2, 0, 3, -3, 0, -1, 1, 4, 0], [-4, 3, 0, -3, 0, -1, 3, 2, 2, 1, 2, 0, 5, -4, -3, 3, -1, 0, -1, -1], [0, -1, 0, 0, 1, 0, 0, -4, -3, 3, 4, -2, 0, 0, 0, 0, 1, -4, -5, 2], [10, -2, -3, 1, 0, 2, -3, 0, 0, -2, 0, 0, -2, 3, 1, -2, 0, 0, 2, 0], [0, -2, 0, 0, 4, 0, 0, -4, -9, 9, 4, -12, 0, -3, 3, 0, 2, -1, -3, 3], [2, 2, -1, -1, 0, 2, -1, -2, 0, -6, -2, 0, 4, -1, -3, 2, -2, 0, 2, -2], [-8, 0, 0, 5, 0, 0, 1, -3, -3, -3, -3, 0, 3, -2, -2, 0, 0, 0, 0, 0], [-4, 5, 3, -1, 0, -3, 5, 6, 2, 3, 6, 0, -4, -1, 2, 5, -1, 0, -3, -1], [0, 0, 3, -3, 0, 2, 1, -2, 2, -2, -2, 0, -3, -1, -3, 0, -1, 0, 2, -1], [-2, 2, 0, -4, 0, -2, 0, 4, 3, 5, 4, 0, -4, -2, 0, 2, 0, 0, -2, 0], [0, -2, 0, 0, -2, 0, 0, 0, -6, 6, 0, 6, 0, 0, 0, 0, 4, -4, -2, 0], [-2, -5, -6, 0, 0, -5, 0, 4, -8, 7, 4, 0, -2, -2, 3, -5, 5, 0, -5, 5], [0, 9, 0, 0, -3, 0, 0, 5, 3, -3, -5, -8, 0, 0, 0, 0, 0, 6, 3, -2], [-6, -4, 1, 5, 0, 0, 1, -2, -4, 0, -2, 0, -2, -1, -1, -4, 2, 0, 0, 2], [0, -2, 0, 0, -4, 0, 0, 5, 4, -4, -5, 0, 0, 4, -4, 0, -1, -2, -1, -3], [4, 0, 0, 0, 0, 0, 2, 0, 2, 2, 0, 0, 2, -2, -2, 0, 0, 0, 0, 0], [0, 3, 0, 0, 1, 0, 0, -7, -5, 5, 7, 4, 0, -1, 1, 0, 2, -5, -4, 3], [0, 0, 0, 0, -4, 0, 0, -1, -4, 4, 1, 2, 0, 0, 0, 0, 2, -6, -6, 2], [0, 0, 1, 3, 0, 4, -1, -4, 0, -8, -4, 0, 2, 1, -3, 0, -2, 0, 4, -2], [0, -3, 0, 0, -1, 0, 0, 1, 6, -6, -1, -8, 0, 3, -3, 0, -4, -7, 2, -5], [0, -3, -1, -8, 0, -3, 2, 2, -4, 5, 2, 0, 1, -3, 0, -3, 3, 0, -3, 3], [4, 5, 0, 1, 0, 1, -1, -4, 2, -5, -4, 0, 1, 0, -1, 5, -3, 0, 1, -3], [0, 10, 0, 0, -6, 0, 0, 2, 12, -12, -2, 0, 0, 4, -4, 0, -4, 4, 9, -5], [0, 2, 2, 0, 0, -8, 0, 2, -7, 7, 2, 0, -3, -4, 4, 2, 3, 0, -8, 3], [14, -3, 3, -4, 0, 5, -2, -5, -1, -8, -5, 0, -1, 3, -2, -3, -1, 0, 5, -1], [0, 4, 0, 0, 4, 0, 0, 1, -8, 8, -1, 6, 0, -2, 2, 0, 2, 8, 0, 2], [4, -5, 4, 1, 0, -3, -1, -2, -6, 5, -2, 0, 0, -4, -1, -5, 4, 0, -3, 4], [0, -8, 0, 0, -2, 0, 0, 1, -5, 5, -1, 2, 0, -1, 1, 0, 3, 1, -4, 2], [0, -11, 0, 0, -1, 0, 0, -3, 1, -1, 3, 2, 0, 3, -3, 0, 2, 1, 4, 1], [0, 5, 0, 0, -1, 0, 0, -1, 5, -5, 1, -2, 0, 2, -2, 0, -2, -4, 1, 2], [-16, 2, 4, -5, 0, 4, 1, -3, 2, 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