
# q-expansion of newform 1470.2.m.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 = 1470
weight = 2
poly_data = [390625, 0, 0, -100000, 1250, 0, 8800, -1440, 2, -288, 352, 0, 2, -32, 0, 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], 1], [[17286250000, -5904875000, -176434375, -2959145625, 988282000, -9130200, 133354345, -99781289, 19988016, -183704, 10867065, -3356569, -30736, -513400, 151945, 23], 263500000], [[-88433203125, 21019390625, 743925000, 14571370000, -4050419125, -20484575, -655014680, 489526928, -71108757, 2515873, -53602520, 13822768, 334507, 2464225, -711160, -12176], 1317500000], [[7283671875, -7983125000, 436350000, -1294480625, 1382043875, -80373000, 61131440, -86582961, 29256259, -1954056, 4897520, -4866801, 200771, -230600, 234160, -7153], 263500000], [[-6183828125, 9931171875, 2208303125, 1662561875, -1656298125, -290692925, -100536475, 97051123, -21020637, -6615837, -6617515, 5695603, 1210867, 417475, -277595, -52781], 1317500000], [[-3498281250, 3362187500, -278928125, 555072500, -565159000, 64163100, -25413395, 36913924, -14275106, 1205964, -2187965, 2002804, -196524, 91100, -94195, 10532], 164687500], [[-2325781250, 1367131250, -182746250, 401156375, -226977250, 35441490, -19619686, 17642267, -6093858, 759218, -1500066, 799959, -110818, 69810, -36198, 5499], 65875000], [[-16529921875, -8980093750, 5544903125, 2922380000, 1529546125, -834999350, -125413435, -18736688, 63233197, -17542518, -9726795, -6082048, 2971213, 543850, 289765, -124784], 1317500000], [[-10391250000, 3063921875, -4479975000, 2128850625, -356214000, 765575875, -118375880, 46597281, -45725264, 15219891, -7551160, 1413601, -2683056, 411075, -35880, 128033], 1317500000], [[-29524375000, 435328125, 7330671875, 4664830000, -45651000, -1280351475, -180286285, 99497552, 62805512, -23549603, -16782845, -596208, 4513448, 759725, 115, -221264], 1317500000], [[4653906250, -3672203125, 2448231250, -823538750, 654882250, -430608425, 46384230, -46480158, 31497002, -8656053, 3173970, -2394398, 1495498, -145225, 116510, -72414], 329375000], [[-448000000, 8384375, 65686250, 73925875, -2887200, -11581465, -3089990, 1584467, 515392, -205673, -265270, 1427, 40928, 12455, -230, -2029], 8500000], [[107188750000, -19156796875, -12093690625, -17288645000, 3498498000, 1997199925, 726564415, -508697448, -32747888, 36024597, 63213055, -10487208, -7558352, -2860075, 550015, 374136], 1317500000], [[-73231250000, -20797375000, 11838115625, 12378994375, 3333950000, -2104753400, -527876935, 85472887, 168261072, -34237368, -43000695, -13513273, 7167888, 2116200, 616665, -372809], 1317500000], [[-4410859375, 36653359375, -14467050000, 244135000, -6401757375, 2424194175, -46705520, 314783384, -226586671, 49110799, -2981040, 23323064, -8294559, 575, -1106320, 377912], 1317500000], [[559140625, -7407050000, 3471705000, -26643375, 1207441825, -590045520, 10534952, -55705439, 50671601, -11953936, 677992, -4520623, 2025921, -10640, 201256, -95903], 263500000]]

hecke_ring_character_values = [[491, [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], [1177, [0, 0, 0, 0, 0, -1, 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, 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, -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, -1, 0, -1, 0, 1, -1, 1, 0, 1, -1, 0, 0, 0, -1], [0, 0, 0, 0, 1, -1, 0, 1, 0, 0, 0, 1, 1, 0, -1, 0], [0, 1, 0, 0, -2, -1, 1, -1, 3, 0, 0, -1, 1, 0, 1, -1], [-4, 0, -1, -1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, -1, -1], [0, -1, 0, 1, 0, -1, -1, 0, 0, 0, 0, 0, 0, 1, 0, 1], [0, 0, 1, 0, -1, 0, 2, 1, 0, 0, -2, 0, 0, 0, 0, -1], [0, 1, 0, 1, -1, 2, 0, 1, 2, 1, 1, 0, 1, 1, -1, 0], [0, 1, -1, -1, -2, -1, -1, -1, 0, -1, 0, -2, 0, 1, 0, 1], [0, 1, -1, 0, 0, 2, -1, 0, 2, 1, 2, 0, 1, 1, 0, 1], [0, 1, 0, 0, 3, 0, 1, 0, -3, 0, 0, 0, 0, 0, 0, -1], [0, -1, 0, -2, 3, 2, -1, 0, -3, 0, -4, 0, 0, -2, 0, 1], [1, 2, -1, 1, -3, 0, 2, -1, 4, 1, -2, -1, 0, 1, 0, -2], [4, 1, 0, 1, 3, 0, 0, 3, -3, -1, -1, 2, 2, -2, 1, 0], [0, -1, 1, -1, 4, 0, -3, -4, -2, -1, 2, 0, 0, 0, 1, -1], [-4, -2, 1, -1, -1, 2, 2, 2, 0, 1, 0, -1, 1, 1, -1, -2], [0, 1, -2, 0, -2, 0, 2, -2, 2, -1, 1, 0, -1, 1, 0, -2], [0, 1, 0, 0, 1, -2, 3, -1, 0, 0, 0, 1, 2, 0, -2, 1], [0, 1, -1, 1, 4, -2, 3, -4, 2, 1, -2, 0, 1, 1, -1, 1], [0, 1, -1, -1, -2, 0, -5, -2, 0, -1, 0, -2, -1, 1, 1, 1], [-4, -2, 2, 0, 3, 0, -2, 3, -3, 2, 0, 4, 1, -1, 0, 2], [-4, 2, -1, -1, -1, -4, 2, 1, 0, 1, -2, 1, 1, -1, 1, -2], [0, -1, 0, -1, 2, 10, 0, -2, 2, -1, -1, 0, -1, -1, 1, 0], [0, -2, 0, -2, 0, -2, 2, 4, 0, 0, 0, 0, 0, 2, 0, -2], [-4, -1, 1, 0, -9, 2, -3, 3, 0, 1, 0, -9, 2, 0, -2, -1], [0, -1, 2, -1, -1, -4, -1, 1, -1, -1, 0, 0, 0, 0, 1, -2], [-1, 1, 0, 1, 6, -2, 1, 4, -10, 0, 0, 4, 0, 1, 0, -1], [0, -1, -1, -1, 0, -2, 1, -4, 0, -1, 0, 0, 1, 1, -1, -1], [0, 0, -1, -2, -1, -2, -9, 1, 3, 0, 9, 0, -1, -1, 2, 1], [4, 0, -3, 3, -3, -1, 0, 1, 0, -3, 0, -3, 0, -3, 0, 0], [4, 0, -1, -1, 4, 0, 1, 4, -4, 0, 1, 2, -2, 2, -1, -1], [0, -1, -1, 1, 8, -4, -5, -8, 1, -1, 4, 0, 2, 2, -1, 1], [0, 1, 0, -3, -1, 0, 4, -1, 1, -1, 3, -2, 2, -2, -3, 0], [-4, -2, -1, -1, 4, -2, -2, -4, 0, 1, 2, -4, -1, -1, -1, 2], [0, 1, -2, 1, -2, 0, 1, 3, -1, 2, 4, 3, -1, 1, -1, -1], [-8, -1, 0, 1, -4, -8, -1, 1, 3, 0, -4, 1, -1, 1, -1, 1], [0, 1, 3, 2, -4, 3, -1, -2, 0, 3, 0, -4, -1, -2, 1, 1], [0, 1, -2, 3, -1, 2, -2, 1, 4, 1, 3, 0, -1, -1, -3, 2], [0, 2, 2, 0, -2, 10, -2, 2, -4, 2, 4, 0, -1, -1, 0, -2], [0, -3, -2, 0, 2, 0, -2, 2, -2, 3, 1, 2, 1, -1, 0, -2], [-5, -1, 3, -3, 0, -3, -1, 6, -6, -3, -6, 6, 1, -3, 1, 1], [3, 1, 0, 1, 0, 0, 3, -10, 0, 0, 0, 0, -2, -1, 2, 1], [4, -4, 1, -1, 2, 0, -1, 2, -2, 4, 3, 3, -1, 1, -1, 1], [4, 1, 1, -2, 3, 0, -1, 3, -3, -1, -2, 6, -2, 2, -2, 1], [0, -1, 3, 0, 3, 4, -7, -1, 0, 3, 0, 3, -4, 0, 4, -1], [-4, -3, 1, -1, 4, -2, -3, 6, -10, -1, 2, 6, -1, -1, -1, 3], [0, 1, 3, 1, -2, 0, 1, -2, 2, -1, 0, -2, 0, 0, 1, 3], [0, -2, 0, -2, 4, 0, -2, 6, -10, 0, -8, 6, 2, -2, 2, 2], [0, 0, -1, 3, -6, 0, -1, 6, 0, 0, 1, 0, 0, 0, -3, 1], [0, 0, -2, -2, -4, 8, -2, 4, -5, 0, 2, 0, 0, 0, 2, 2], [0, -1, 1, 1, 2, 2, -7, -2, -4, -1, 6, 0, -3, -3, -1, -1], [8, 0, 1, 1, -1, 5, 0, -4, 5, -1, -2, -4, 2, 1, 2, 0], [4, -2, -1, 2, -4, 3, -2, 8, -4, 1, -6, 8, -1, 2, -1, 2], [-8, -1, -1, -3, 6, 0, 1, 6, -6, 1, 2, 8, 0, 0, -3, -1], [0, 1, 0, 2, 6, -4, -4, -6, -3, 1, 5, 0, 0, 0, -2, 0], [-7, 4, 1, -3, 3, 7, -4, 6, 0, 1, 0, 3, -3, 3, 3, 4], [0, 0, 3, 3, 0, 0, -6, 0, 0, 0, -6, -2, -2, 2, 3, 3], [0, -2, 3, 0, 6, 1, -2, -2, 0, 3, 0, 6, -1, 0, 1, -2], [0, 2, 4, 0, -6, -1, -2, 6, 0, 4, 0, -6, 1, 0, -1, 2], [-4, -4, 2, 2, -2, -2, -4, 6, -4, -2, -8, 6, -4, 2, -4, 4], [0, -3, 3, 0, 1, 8, 3, -1, 5, -3, -6, 0, 0, 0, 0, -3], [4, 3, -2, 3, 0, -4, -7, 10, 0, -2, 0, 0, 3, -3, -3, 3], [-5, -1, -4, 1, -10, 4, 5, 6, 0, -4, 0, -10, 2, -1, -2, -1], [4, 1, 1, 6, 3, 0, 7, 3, -3, -1, 6, 8, 0, 0, 6, 1], [0, 3, 2, -3, 2, -1, -3, -8, 0, 2, 0, 2, -2, 3, 2, 3], [8, 0, 0, 2, -6, -6, -12, 0, 0, 0, 0, -6, 0, -2, 0, 0], [8, 0, 2, 4, 6, 0, 2, 6, -6, 0, 2, 4, 3, -3, 4, 2], [-4, 0, -3, 0, -9, 3, 8, 2, 0, -3, 0, -9, 1, 0, -1, 0], [0, -1, 2, -4, 6, 2, -6, -6, 10, -1, 5, 0, 3, 3, 4, -2], [-4, 1, 1, 0, 3, -6, 1, -1, -2, -1, 6, -1, 2, 0, 2, -1], [-5, 0, 3, -1, 5, -3, 0, 12, -17, -3, -6, 12, -1, -1, -1, 0], [0, -3, -3, -2, 5, -8, -7, -5, -1, -3, 4, 0, -2, -2, 2, 3], [4, 2, -3, -1, -1, -4, -6, -8, 0, -3, 0, -1, -1, 1, 1, 2], [0, -1, -4, -2, 6, 8, -5, -6, -1, -1, 4, 0, 0, 0, 2, 4], [4, 2, 1, 1, -12, 2, 2, 4, 8, -1, -2, 4, 1, 1, 1, -2], [8, 1, 2, 2, -4, 0, -5, -4, 4, -1, -6, 2, 3, -3, 2, 2], [-8, 2, -2, -2, -1, 0, 2, -1, 1, -2, 0, 7, 2, -2, -2, -2], [4, -2, 2, 2, 0, 0, -10, 0, 0, 2, -8, 8, -2, 2, 2, 2], [4, 3, -1, -2, 5, 0, 9, 5, -5, -3, 6, 0, 4, -4, -2, -1], [-8, 2, -1, -1, -4, 0, 5, -4, 4, -2, 3, 4, 2, -2, -1, -1], [-4, 3, 0, 1, 5, 4, 1, 9, 0, 0, 0, 5, 1, -1, -1, 3], [-12, 2, -3, -2, -9, 0, 3, -9, 9, -2, 1, -15, 0, 0, -2, -3], [-4, 4, -2, 0, -4, -8, 4, 2, 2, 2, 0, 2, 4, 0, 4, -4], [0, 2, -4, -2, -2, 8, -4, 2, 10, 2, 6, 0, 0, 0, 2, 4], [9, 1, 1, 3, -14, -5, -1, 4, 0, 1, 0, -14, -1, -3, 1, 1], [0, -1, 1, -1, 4, 0, -7, -4, 8, -1, 6, 0, 4, 4, 1, -1], [-8, 0, 0, -6, 10, 0, 0, -2, -8, 0, 0, -2, -2, -6, -2, 0], [4, 0, -6, 0, 4, 4, 0, -6, 2, 6, 8, -6, 0, 0, 0, 0], [0, 1, 1, 2, -9, 0, 3, -9, 9, -1, 2, -8, -4, 4, 2, 1], [0, 2, 2, 4, -2, -2, -2, 4, 0, 2, 0, -2, 6, -4, -6, 2], [-8, -2, 3, -1, 8, 0, -3, 8, -8, 2, -1, 9, 5, -5, -1, 3], [0, 5, -1, 2, 7, 8, 3, -7, 11, 5, 2, 0, -2, -2, -2, 1], [-8, -4, -3, 2, -6, 8, 0, -1, 0, -3, 0, -6, 2, -2, -2, -4], [0, 4, -1, 0, 10, 0, 1, 10, -10, -4, -3, 8, -2, 2, 0, -1], [0, 4, -4, 1, 7, -4, 4, -7, 1, 4, 0, 0, 0, 0, -1, 4], [0, -1, 0, 1, 6, 3, -3, 2, 0, 0, 0, 6, -2, -1, 2, -1], [-12, -1, -3, -3, -8, 0, 7, -8, 8, 1, 8, -12, -5, 5, -3, -3], [-12, 2, 1, 0, 2, 10, -2, -9, 0, 1, 0, 2, 2, 0, -2, 2], [-1, 1, 0, -7, 0, -4, 3, -2, 0, 0, 0, 0, -2, 7, 2, 1], [8, 3, -1, 1, -6, -6, -7, 6, 0, -1, 0, -6, -1, -1, 1, 3], [0, 0, -3, 3, 0, 12, -10, 0, 2, 0, 10, 0, -2, -2, -3, 3], [20, -2, 0, -4, -4, 0, 0, -4, 4, 2, 2, -4, 2, -2, -4, 0], [8, 2, -5, 0, -2, 12, 2, -8, 10, 5, 6, -8, -4, 0, -4, -2], [0, 1, -5, -3, 2, 2, 1, -2, 0, 5, 6, -2, 1, -3, 1, -1], [8, -3, 0, 0, 0, -11, -5, -13, 0, 0, 0, 0, 3, 0, -3, -3], [0, 0, 3, 3, -2, 8, -1, 2, 4, 0, 1, 0, 0, 0, -3, -3], [0, 4, -2, -4, 3, 0, -10, -3, 0, 4, 14, 0, 4, 4, 4, 2], [8, -1, -3, 0, 5, 10, -1, 7, -12, 3, 6, 7, -2, 0, -2, 1], [0, -1, 5, 7, 6, -5, -1, 7, -13, -5, -12, 7, -2, 7, -2, 1], [-4, -5, 2, 0, -1, 3, 3, -5, 0, 2, 0, -1, 1, 0, -1, -5], [-4, -1, 2, 8, -6, 0, -2, -6, 6, 1, -1, -4, 1, -1, 8, 2], [8, -5, 0, -1, 9, 0, -8, 9, -9, 5, -3, 14, -2, 2, -1, 0], [-16, 0, -1, -5, 4, 0, -4, 4, -4, 0, -4, 8, 0, 0, -5, -1], [-8, -1, -4, -1, 0, 3, -3, 4, 0, -4, 0, 0, 4, 1, -4, -1], [8, 1, 0, 6, 1, 4, 1, 14, -15, 0, 4, 14, -2, 6, -2, -1], [-4, 0, 2, 5, -5, -2, 0, 9, -4, -2, -14, 9, -7, 5, -7, 0], [0, -1, 6, 2, -8, 20, 6, 8, 5, -1, -7, 0, -2, -2, -2, -6], [0, 2, -2, 1, -8, -8, 2, 8, -6, 2, 0, 0, 0, 0, -1, 2], [0, 5, 0, 1, -10, -7, 5, 2, 8, 0, 6, 2, 6, 1, 6, -5], [0, 1, 3, 4, 4, -1, 1, 2, -6, -3, -6, 2, -3, 4, -3, -1], [-8, 2, -2, -6, 8, 6, 2, 2, -10, 2, 8, 2, -8, -6, -8, -2], [0, -5, 0, 2, 10, 10, 4, -10, -4, -5, -9, 0, -3, -3, -2, 0], [-8, -2, -5, 2, 9, 0, 4, 9, -9, 2, 6, 7, 0, 0, 2, -5], [0, -7, 2, 5, 1, 0, 3, -1, -5, -7, -10, 0, 2, 2, -5, -2], [-8, -5, -1, 0, -9, 0, -7, -9, 9, 5, -2, -8, -2, 2, 0, -1], [4, 1, -7, 0, -1, 4, 1, -3, 4, 7, 6, -3, 0, 0, 0, -1], [12, 2, 1, 2, -8, -8, -2, 6, 0, 1, 0, -8, -2, -2, 2, 2], [0, -1, 1, 4, -1, -8, 9, 1, 1, -1, -10, 0, 2, 2, -4, -1], [8, -4, 2, -2, 6, 6, -4, -6, 0, -2, -4, -6, 4, -2, 4, 4], [-16, -1, -4, -3, 7, 0, 16, 7, -7, 1, 17, 2, 0, 0, -3, -4], [4, -2, -2, 6, 4, 7, 4, -7, 0, -2, 0, 4, -5, -6, 5, -2], [0, -5, -1, 2, -8, -2, -1, 8, 2, -5, -4, 0, 7, 7, -2, 1], [0, 1, -3, 0, 10, 0, -1, 10, -10, -1, -2, 17, 2, -2, 0, -3], [-8, 1, 5, -5, -6, 5, -1, 14, 0, 5, 0, -6, -2, 5, 2, 1], [4, 2, -2, 2, -16, 0, 2, -4, 20, 2, 8, -4, 2, 2, 2, -2], [0, -2, -2, -1, 2, -5, -2, -16, 14, 2, 4, -16, 6, -1, 6, 2], [0, 0, 6, -6, 8, -16, -2, -8, 12, 0, 2, 0, 4, 4, 6, -6], [0, 0, -5, 3, 10, -20, 3, -10, -8, 0, -3, 0, 4, 4, -3, 5], [0, 4, 2, 0, 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