
# q-expansion of newform 810.3.j.f, downloaded from the LMFDB on 18 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 = 810
weight = 3
poly_data = [961, 0, -434, 0, 165, 0, -14, 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], 1], [[721, 0, 0, 0, 0, 0, 1, 0], 495], [[0, 1711, 0, 0, 0, 0, 0, 1], 495], [[0, 721, 0, 0, 0, 0, 0, 1], 495], [[6076, 0, -2310, 0, 165, 0, -14, 0], 5115], [[29078, 0, -11055, 0, 1155, 0, -67, 0], 15345], [[0, 29078, 0, -11055, 0, 1155, 0, -67], 15345], [[0, 12493, 0, -24915, 0, 2145, 0, -182], 15345]]

hecke_ring_character_values = [[731, [0, 0, 0, 0, 1, 0, 0, 0]], [487, [-1, 0, 0, 0, 0, 0, 0, 0]]]
aps_data = [[0, 1, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 3, 1, 0, 0], [0, 0, -2, 0, 0, 0, -1, -2], [0, 0, 3, 0, 0, 0, -2, 3], [0, 0, -3, 0, 0, 0, 0, 0], [3, -7, 0, 0, 0, 0, 0, 0], [-3, -3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 15, 14, 0, 0], [0, 0, 0, 0, 0, 0, -9, 0], [0, 0, 0, 0, 17, 15, 0, 0], [0, 0, 0, 14, 0, 0, -14, 2], [0, 0, -6, -1, 0, 0, 0, 0], [0, 0, -4, 0, 0, 0, 7, -4], [-60, 14, 0, 0, 60, -14, 0, 0], [-51, 9, 0, 0, 0, 0, 0, 0], [0, 0, 9, -4, 0, 0, 0, 0], [49, -24, 0, 0, -49, 24, 0, 0], [0, 0, -26, 8, 0, 0, 0, 0], [0, 0, 0, 6, 0, 0, -6, 21], [0, 0, 0, 16, 0, 0, -16, 7], [-45, 63, 0, 0, 45, -63, 0, 0], [27, -58, 0, 0, -27, 58, 0, 0], [0, 0, 0, 29, 0, 0, -29, 0], [0, 0, -22, 0, 0, 0, 10, -22], [0, 0, -6, 0, 0, 0, 8, -6], [0, 0, 2, 28, 0, 0, 0, 0], [36, 30, 0, 0, 0, 0, 0, 0], [-19, -90, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 12, 30, 0, 0], [0, 0, 0, -34, 0, 0, 34, 14], [0, 0, 12, 18, 0, 0, 0, 0], [27, 135, 0, 0, -27, -135, 0, 0], [0, 0, 0, 0, -38, 102, 0, 0], [0, 0, -12, 35, 0, 0, 0, 0], [20, -66, 0, 0, -20, 66, 0, 0], [0, 0, 19, -16, 0, 0, 0, 0], [0, 0, 0, -28, 0, 0, 28, -10], [0, 0, 0, 0, 21, -60, 0, 0], [69, -29, 0, 0, -69, 29, 0, 0], [0, 0, 0, -20, 0, 0, 20, -12], [113, -42, 0, 0, 0, 0, 0, 0], [0, 0, -12, 0, 0, 0, -66, -12], [0, 0, -17, -34, 0, 0, 0, 0], [-279, 1, 0, 0, 0, 0, 0, 0], [114, -30, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -171, -105, 0, 0], [0, 0, 2, 0, 0, 0, -74, 2], [-87, 232, 0, 0, 87, -232, 0, 0], [0, 0, 0, 0, 59, -114, 0, 0], [-126, -82, 0, 0, 0, 0, 0, 0], [0, 0, -48, -44, 0, 0, 0, 0], [191, 90, 0, 0, -191, -90, 0, 0], [0, 0, 0, 36, 0, 0, -36, 30], [0, 0, 0, 0, 87, -213, 0, 0], [60, -118, 0, 0, -60, 118, 0, 0], [0, 0, 0, 18, 0, 0, -18, 72], [115, -147, 0, 0, 0, 0, 0, 0], [0, 0, 49, 0, 0, 0, -22, 49], [0, 0, 0, 0, 0, 0, 27, 0], [0, 0, 86, -53, 0, 0, 0, 0], [0, 0, 0, 0, -45, -129, 0, 0], [0, 0, 0, 0, 0, 0, 0, 60], [0, 0, -21, -114, 0, 0, 0, 0], [0, 0, 20, 0, 0, 0, 112, 20], [57, 211, 0, 0, -57, -211, 0, 0], [266, 126, 0, 0, -266, -126, 0, 0], [0, 0, 95, -56, 0, 0, 0, 0], [0, 0, 0, 0, -150, -240, 0, 0], [37, -84, 0, 0, -37, 84, 0, 0], [6, -182, 0, 0, -6, 182, 0, 0], [0, 0, 0, 18, 0, 0, -18, -84], [0, 0, 16, 0, 0, 0, 107, 16], [0, 0, -12, -18, 0, 0, 0, 0], [91, 165, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 465, 76, 0, 0], [0, 0, 78, 0, 0, 0, 119, 78], [0, 0, 0, -106, 0, 0, 106, 23], [0, 0, 18, 106, 0, 0, 0, 0], [0, 0, 0, 0, 491, 30, 0, 0], [0, 0, 63, -50, 0, 0, 0, 0], [-391, -24, 0, 0, 391, 24, 0, 0], [0, 0, 0, -152, 0, 0, 152, 33], [0, 0, 0, -8, 0, 0, 8, 103], [-135, -189, 0, 0, 135, 189, 0, 0], [447, -6, 0, 0, -447, 6, 0, 0], [0, 0, 0, -33, 0, 0, 33, -132], [0, 0, 82, 0, 0, 0, 98, 82], [0, 0, -30, 0, 0, 0, 78, -30], [0, 0, -96, 132, 0, 0, 0, 0], [675, 48, 0, 0, 0, 0, 0, 0], [0, 0, 93, 0, 0, 0, -34, 93], [0, 0, 0, 51, 0, 0, -51, 30], [0, 0, -132, -18, 0, 0, 0, 0], [0, 0, 0, 0, 681, -57, 0, 0], [357, 54, 0, 0, 0, 0, 0, 0], [0, 0, -192, 54, 0, 0, 0, 0], [0, 0, 0, 17, 0, 0, -17, 6], [0, 0, 0, 137, 0, 0, -137, 86], [-80, 210, 0, 0, 0, 0, 0, 0], [0, 0, 10, 0, 0, 0, -67, 10], [-210, 66, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 384, 106, 0, 0], [0, 0, -12, 0, 0, 0, -19, -12], [0, 0, 0, 0, -127, 699, 0, 0], [0, 0, 0, 170, 0, 0, -170, 77], [-309, -162, 0, 0, 309, 162, 0, 0], [549, 237, 0, 0, 0, 0, 0, 0], [0, 0, 75, -162, 0, 0, 0, 0], [-437, 30, 0, 0, 437, -30, 0, 0], [0, 0, 16, 53, 0, 0, 0, 0], [0, 0, 0, -176, 0, 0, 176, -14], [0, 0, 0, 0, 315, -269, 0, 0], [266, -174, 0, 0, -266, 174, 0, 0], [-427, 153, 0, 0, 0, 0, 0, 0], [0, 0, -192, 0, 0, 0, -38, -192], [0, 0, 12, -45, 0, 0, 0, 0], [339, 252, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -855, -49, 0, 0], [0, 0, -27, 0, 0, 0, 94, -27], [0, 0, 0, 0, -668, -258, 0, 0], [0, 0, 33, 0, 0, 0, -84, 33], [444, 94, 0, 0, -444, -94, 0, 0], [147, 268, 0, 0, 0, 0, 0, 0], [65, -345, 0, 0, -65, 345, 0, 0], [0, 0, 0, 100, 0, 0, -100, -114], [-636, 378, 0, 0, 636, -378, 0, 0], [0, 0, 0, -28, 0, 0, 28, -18], [0, 0, 108, 0, 0, 0, -36, 108], [0, 0, 92, 334, 0, 0, 0, 0], [827, -51, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 498, 576, 0, 0], [0, 0, 0, 0, -473, -537, 0, 0], [0, 0, 0, -166, 0, 0, 166, 2], [0, 0, -156, 153, 0, 0, 0, 0], [0, 0, 0, 0, 687, -102, 0, 0], [459, 45, 0, 0, 0, 0, 0, 0], [0, 0, -82, -155, 0, 0, 0, 0], [0, 0, 0, 0, 1011, -281, 0, 0], [0, 0, 0, 15, 0, 0, -15, -90], [-692, 282, 0, 0, 0, 0, 0, 0], [0, 0, 192, 0, 0, 0, -100, 192], [0, 0, 114, -189, 0, 0, 0, 0], [615, -424, 0, 0, 0, 0, 0, 0], [180, -930, 0, 0, 0, 0, 0, 0], [0, 0, 72, 0, 0, 0, -68, 72], [0, 0, 40, 0, 0, 0, -88, 40], [945, -537, 0, 0, -945, 537, 0, 0], [0, 0, 0, 0, 653, 315, 0, 0], [-75, 30, 0, 0, 0, 0, 0, 0], [0, 0, -83, -226, 0, 0, 0, 0], [0, 0, 0, 117, 0, 0, -117, 342], [0, 0, 0, -123, 0, 0, 123, -90], [0, 0, 0, 0, 531, 798, 0, 0], [0, 0, -102, 0, 0, 0, 318, -102], [0, 0, -111, 0, 0, 0, -348, -111], [-762, -366, 0, 0, 0, 0, 0, 0], [0, 0, 120, 0, 0, 0, 129, 120], [0, 0, 0, -14, 0, 0, 14, -176], [0, 0, -180, -4, 0, 0, 0, 0], [399, -150, 0, 0, -399, 150, 0, 0], [294, -814, 0, 0, 0, 0, 0, 0], [0, 0, 274, -136, 0, 0, 0, 0], [0, 0, 0, -342, 0, 0, 342, 90], [0, 0, 0, 0, -678, 402, 0, 0], [243, 70, 0, 0, -243, -70, 0, 0], [433, -207, 0, 0, 0, 0, 0, 0], [0, 0, 159, 0, 0, 0, -120, 159]]
