
# q-expansion of newform 8046.2.a.m, 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 = 8046
weight = 2
poly_data = [-375, 910, 761, -2249, -417, 2013, -63, -774, 106, 116, -21, -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], 1], [[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 1], [[1592878652, -3729321842, -2256579470, 6463643167, 447932378, -3495027303, 410288818, 609086107, -109537587, -27408721, 5673372, -32197], 5855999], [[-4301878920, 9660687071, 6698494966, -16685502287, -2097844062, 8970420017, -664646379, -1586491494, 222792211, 77892894, -12051765, -337779], 29279995], [[1309434950, -5435421687, 2382541943, 8040696879, -6124911496, -3268300794, 3479105328, 137663433, -610246892, 68317507, 27982820, -4626967], 29279995], [[2800717475, -8170221548, 594551152, 13075477756, -6551037399, -6005098721, 4419720832, 526825367, -818338273, 69519048, 38266875, -5834388], 29279995], [[-6906022130, 12966000711, 17416329301, -22956311877, -15262968602, 13696849342, 5470397091, -3330885579, -795142464, 329349654, 33647510, -11071809], 29279995], [[-1289750025, -1591143977, 13154737748, 804575664, -19399390581, 1916697621, 9679352383, -1697104197, -1640249962, 327537312, 74241930, -15600107], 29279995], [[-6112777815, 18959758726, -3005054784, -30654436132, 17633361118, 14014610012, -11652307114, -1047579084, 2153825106, -207068641, -100688050, 15915726], 29279995], [[-38973036, -842422122, 2366997820, 1171309316, -3967835268, -179845605, 2107854700, -254489192, -366093260, 64707528, 16732001, -3317302], 5855999], [[5205009495, -6697290927, -19695978447, 13297311074, 22410652829, -9615020874, -9796137762, 3159684708, 1573561298, -418005468, -69687500, 17078923], 29279995], [[-395658693, 2627858133, -3278532055, -3951467976, 6428622786, 1341737333, -3586733161, 238661630, 632612308, -98087514, -29080114, 5445786], 5855999]]

hecke_ring_character_values = None
aps_data = [[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, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1], [-1, 0, 0, 0, -1, 0, 1, 0, -1, -1, -1, 0], [-1, 0, 2, -1, 0, 0, 1, -1, 0, -1, 0, -1], [-2, 0, 0, 1, 0, -1, -1, 0, -1, 1, 0, -1], [-2, 1, 0, 1, 0, 0, -1, 0, 0, 1, 1, 0], [-3, -2, 0, 0, -2, 0, -1, 0, 0, 0, -1, -1], [-1, 2, 0, 0, 1, 2, -1, 1, 1, -1, 0, 0], [-2, 1, 0, 0, 0, 1, -1, 1, -1, 0, 0, 1], [-2, 1, 2, -1, 0, 1, -1, 0, 1, 1, 0, 0], [-1, 1, 1, 0, 1, 1, 1, 0, 0, 1, -1, 1], [0, 1, 0, 0, 2, 1, -1, 1, 2, 1, 1, 1], [-2, 0, 0, -1, -1, 0, -1, 2, -1, -1, 0, -2], [0, 0, 1, 0, 2, -2, 3, -2, -1, -1, 0, 2], [-2, 1, -2, 1, -3, 0, -1, 1, 1, -1, 1, 0], [-3, 0, 0, 1, 0, -1, 0, 0, -1, 0, -2, -1], [-3, 1, 1, 1, 2, 1, -1, -2, 1, 1, -1, 2], [-1, 0, 1, 0, -1, 1, 2, 1, 1, -1, 1, -1], [0, 0, -1, -1, 0, 1, 1, -2, -1, -1, 0, 0], [0, 0, -1, 0, 1, -2, 0, 0, 1, 1, 0, 1], [-2, -3, 2, -2, -1, -1, 4, 0, -1, 0, 0, -3], [-1, 1, 1, 0, 1, 1, 2, -1, 1, 1, 0, 2], [-6, -1, -2, 2, 0, 0, 2, 2, -2, -1, 1, 0], [1, 0, -2, -1, 1, 0, -1, -1, 2, 0, 1, 1], [-4, -3, -1, 2, -3, -2, 3, -1, -2, -3, 0, -1], [2, -2, 1, -2, 0, -2, 0, 0, -1, -1, 0, -2], [-1, 0, 2, 0, 1, -1, -1, -2, 0, -1, -1, 1], [1, -2, -2, 0, -1, 1, 1, -3, -1, -2, -1, 1], [0, -1, -4, 1, -2, -4, 1, -2, -4, 0, -1, 2], [-3, -3, -2, 1, -3, -5, 0, -1, -1, -1, -2, -1], [-1, 2, -2, 0, 1, -1, -4, 2, 1, 2, 1, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 0, -4, 0, -3, -3, 0, 2, 0, -4, 2, -1], [0, 3, -5, 1, 0, 0, -2, 2, 2, 0, 4, 2], [-2, -2, 0, 1, 0, 1, 0, 3, 2, -1, 3, -2], [3, -2, -1, -1, 1, 0, -1, -1, -2, 5, 2, 0], [1, -6, 2, -3, 1, -4, 3, -5, -2, 1, -2, -1], [-3, -3, 1, 0, 0, -3, 4, 0, 1, 1, 0, -1], [8, 0, 3, -1, 3, 4, 3, -2, 2, -1, -2, 3], [-7, 2, -1, 0, 0, 4, 2, 4, -1, -2, 1, 1], [3, 1, -1, -1, 1, -2, 1, -3, -1, 0, -3, 2], [-9, 1, -2, 2, -1, 5, -3, 4, 1, -1, 1, 2], [2, 2, -1, 1, 1, 1, -6, 4, 2, 5, 3, 1], [4, -1, -1, -1, 1, 2, 2, -1, 2, 0, 2, 3], [-3, 1, -3, 2, 1, -3, -5, 3, 2, 3, 4, -1], [-2, -2, -1, -2, -1, 4, 0, -1, 1, -4, -2, 1], [-7, 2, -2, 4, -3, 6, -3, 4, -2, -3, -4, 2], [-7, -4, 3, -1, 1, 2, 5, -3, -2, -3, -1, 0], [-1, -3, -2, 0, -6, -2, 1, -1, -1, 1, -1, -5], [3, 3, 1, 0, 2, 5, -2, 3, 3, -3, -1, 4], [-7, 5, -2, -1, 1, 3, -5, 7, 3, 3, 2, 1], [2, 9, 1, 2, 6, 1, -2, 0, 1, 4, 1, 2], [1, 0, -2, 1, 2, -2, 2, 0, 4, 1, 4, -1], [-1, -4, -5, 1, -2, -3, 3, 0, -4, -5, -5, 2], [4, 1, 1, -2, 1, 7, -2, 2, 2, -4, -3, 0], [-1, 2, 5, 0, 4, 2, 1, 0, 0, 7, 3, 1], [-4, 0, 0, -2, -1, -4, -2, -1, 0, -1, 0, -3], [-1, -2, -1, 0, 1, -1, -2, -2, -1, 5, 4, 0], [2, -8, -5, 1, -4, -4, 1, -1, -3, -4, -1, -4], [1, -8, -2, 0, -9, -5, 2, 0, -3, -4, -2, -3], [2, 1, -3, 2, 0, 2, -3, 0, 4, -2, -2, 3], [1, -5, -4, 0, -2, -1, -2, 1, 1, -6, -3, 0], [-6, 0, -3, 4, -2, -2, -5, 2, -1, -1, -4, 0], [-2, 1, 4, 0, 2, -1, 0, 0, 0, -1, -1, -3], [-2, 2, 3, -1, -1, 1, -7, 6, 4, 2, 0, -4], [5, -1, -3, -1, 0, 1, -3, 2, -1, 4, -2, -1], [-1, 4, -2, 3, -1, 2, -1, 4, 2, 0, 3, 0], [-5, -2, 4, -1, 2, 0, 0, -3, -4, 5, 2, -3], [-1, 2, 1, -2, 4, -3, 1, -7, -1, 1, -4, 4], [-5, 0, 1, 1, 3, -2, 2, -3, -3, 0, 0, 2], [3, -3, 5, -3, -2, 0, 3, -3, 0, -3, -5, -1], [6, -3, 0, 3, 1, -2, 1, 0, 1, 3, -1, 2], [-3, -2, 2, 3, -1, -5, 3, -7, -3, -1, -4, 1], [-7, -1, 5, -1, 1, 3, -2, 3, 0, -3, -3, -5], [2, -3, -3, 3, -7, 2, 0, 2, 2, -2, 0, 1], [0, 4, -2, 3, 0, 4, 3, -3, -4, -4, -2, 7], [2, -4, 2, 1, -4, -1, 1, -5, -3, 1, -1, -4], [-7, 3, 1, 2, -1, -1, 2, 3, -1, 1, 1, -1], [-4, -1, 5, -3, 2, 0, 1, -2, -4, 3, 3, -2], [-5, 2, 3, 1, 4, -2, 1, 4, 2, 4, 7, -3], [7, -4, -2, 3, 2, 0, 1, -5, -1, 0, -1, 0], [2, 3, 3, -2, 7, 0, -1, -2, 2, 1, 2, 0], [2, 2, 6, 0, 7, 0, 2, -4, 4, 4, 5, 0], [-6, 2, 0, 0, 3, -1, 2, 2, -3, -1, 2, -1], [3, 1, 4, 0, 4, -2, -2, 0, 4, 1, 3, -2], [1, 0, -4, -1, 1, 0, -1, 3, 1, 5, 4, 1], [-4, 2, -1, 0, 1, 6, 1, 7, 2, -3, 2, 3], [-3, 9, 5, -3, 5, 1, -3, 0, 4, 2, -1, 2], [-5, -5, -2, 2, -5, -5, 1, 0, -7, 3, 0, -4], [6, -4, 0, -3, 0, -7, 3, 2, -3, 1, -1, -6], [2, -4, -1, -3, -2, 3, 1, -4, -1, -1, -3, 1], [-2, -2, -1, 3, -1, -4, 1, -2, -6, -3, -4, -3], [-4, 2, 8, 0, 5, -3, -2, 0, 3, 5, -1, -1], [1, -1, 4, -1, -2, 6, -1, 2, 2, 2, -5, 0], [-1, 9, 5, -4, 7, 6, 0, -1, 5, 3, 3, 2], [1, -9, 2, -3, -5, -4, 0, -1, -1, 5, -2, -6], [8, -5, 2, -3, 1, -3, 4, -6, -4, 1, -3, 0], [-7, 2, -1, 2, 0, 2, 0, 1, -2, 2, 2, 4], [-7, 0, -5, 4, -5, 1, -3, 0, 2, -1, 2, 2], [1, 5, 4, -3, 4, 4, -4, 1, 0, 1, -6, 4], [-14, -1, 5, 0, -4, -5, 5, -2, -7, -4, -6, -8], [8, -7, 0, 0, -3, -1, 8, -4, -6, -4, -4, 3], [4, 3, -3, 1, 0, -4, 3, -3, -4, 2, 5, 5], [9, 2, 0, 0, 6, -4, -1, -3, 5, 5, 4, 6], [-5, 5, 3, -2, -6, -1, -6, 1, 4, 0, 0, -5], [-5, -6, -4, 0, -10, 2, -1, 0, -4, -7, 2, -9], [10, -1, 3, -5, -1, -2, 0, -1, 0, 0, -1, 1], [0, 2, -4, -2, -4, 9, 0, 9, 2, -6, -2, 2], [7, 3, 3, 1, 7, 2, 3, -3, 1, 0, -2, 3], [6, -7, -3, -3, -10, -5, -1, 3, -7, -4, -4, -6], [2, -2, 2, -6, -1, 2, -4, -1, 1, -1, -1, 1], [0, 6, -8, 5, -3, 5, -5, 3, 1, -6, 0, 5], [-3, 2, 0, 0, 6, 0, 0, -4, 5, 3, -4, 6], [0, 2, -3, 2, 2, 5, -1, 5, 0, 1, 2, 1], [5, -6, 8, -7, -2, -5, 6, -3, -2, -3, -3, -3], [2, 3, 7, -3, 6, 2, -6, -1, 3, 11, 2, -1], [-7, 1, 0, 1, -6, 4, -3, 5, -1, 1, -4, -6], [8, 2, 4, 1, -2, 5, 2, -3, -5, -2, 2, -2], [8, -5, 0, -2, 0, -5, 3, 1, -2, 4, 2, -3], [3, 4, 3, 0, 4, -1, 6, -4, 2, 9, 3, 4], [-6, 6, 3, -1, 7, 5, -1, 2, 7, 2, 2, 2], [-14, -1, 0, 0, -5, 0, 4, -3, -3, -3, -4, 2], [-14, -1, -3, 3, -9, 0, -7, -1, -3, -2, -4, -4], [-5, -2, -2, -1, -9, 4, -1, 2, 1, -5, 0, -1], [-13, -1, 6, -5, -7, -2, -3, 4, -1, 1, 0, -5], [-3, 7, -2, 2, -2, 0, -5, 9, 4, 2, 4, 2], [3, 2, -3, 4, -4, -3, 0, 4, -4, -3, 0, -3], [8, 5, -2, 2, 0, -1, -5, -2, 4, -1, 0, 5], [-3, 0, 3, 0, -1, 1, 6, 0, 3, -1, 6, -3], [19, -6, -2, -3, 3, -7, 7, -9, 3, 2, 1, 1], [9, 1, -2, 2, 5, 3, -5, 0, 0, 10, 4, 4], [6, 2, -3, 5, 7, -3, 4, -2, -2, 2, 1, 6], [5, 7, 7, 3, 5, 4, 1, 0, 4, 7, 7, 2], [4, -7, 2, -2, 1, -6, 2, 1, 5, 5, 4, -1], [2, 7, -2, 5, 4, 6, -5, -5, 7, 2, 3, 10], [6, -3, -2, 0, -1, -3, -8, 5, 1, 6, 2, 0], [-4, 4, -6, 4, 2, 0, -6, 2, -3, 6, 6, 3], [4, 3, -3, 2, -7, -1, -5, 7, -1, 3, 4, -10], [-3, 0, -7, -3, -2, 0, -2, 3, 7, 3, 7, 0], [3, 5, -1, 0, 2, 6, -4, -1, 7, -2, 2, 8], [-3, 2, 0, 2, 6, -3, 4, -1, -4, 0, 0, 3], [9, 4, 4, -1, 4, -2, 3, -4, -1, 5, 3, -3], [2, 0, 1, -6, -3, -1, -1, -1, 0, -2, -5, 1], [-10, -3, -4, -2, -7, 4, -6, 4, -1, -4, -1, -10], [-13, 2, 6, -2, -6, 4, -1, 6, 2, -1, -2, -4], [5, 3, -5, 6, 8, -3, 3, 3, -1, 7, 6, 6], [4, -2, 1, -6, 1, 4, -8, 3, 2, 3, 2, 0], [3, -4, 3, -1, -2, -2, -8, 1, -3, 1, -3, -12], [-15, 2, 1, 4, -2, 2, -7, -1, 0, 5, 1, -8], [14, 9, 1, 4, 9, 5, -1, 3, 5, -3, 0, 8], [5, 5, -2, 7, 1, 0, -4, 1, 5, 6, 5, -2], [-14, -3, 7, -2, 1, -7, 5, -5, 2, 1, -3, -2], [3, -1, 7, -7, 4, -4, 3, -4, 2, 3, 1, -7], [12, -11, -8, -4, -4, 3, 2, 4, -5, 0, -4, 0], [9, -5, 5, -8, 1, -6, 10, -4, -2, -2, 0, -6], [21, -1, -4, 2, 0, -2, 4, 1, -3, -1, -2, 6], [11, -8, -1, -4, 3, -8, 12, -10, -2, 4, 6, -1], [3, -3, -5, 1, -7, -3, 2, 0, -2, 2, 6, -4], [-3, 0, -13, 6, 3, -4, -3, 3, 0, 0, 2, 1], [3, -3, 7, -4, 9, 5, 4, -5, -2, 7, -1, 4], [10, -2, 8, -3, 3, 4, 12, -5, 0, -7, -6, 2], [9, -4, 5, -4, 0, 6, -2, -3, 7, -3, 1, -1], [7, 4, -8, 0, -5, 7, -6, 4, 2, -14, -4, 2], [1, 9, 2, -4, 0, 8, -6, 7, 3, -3, -2, 3], [-3, -3, 11, -3, -1, -4, 4, -3, 3, -4, -2, -3], [10, 2, -3, 5, 1, -5, -1, -5, 0, 0, 1, 5], [9, 10, -5, 2, 12, 2, -2, 1, 7, 11, 5, 9], [4, 2, -7, 2, 6, -1, -7, 1, 3, 5, 9, -2], [-3, 1, -2, 3, -5, -6, -4, -2, -7, 5, 2, -7], [-19, 6, 4, 5, 1, 3, 0, 1, -9, 1, -1, -3], [-10, 5, 0, 1, 3, 9, -9, 5, 5, 2, -4, -4], [10, -9, 9, -8, 2, -3, 1, -5, 12, 0, 2, -5], [-6, 10, -8, 11, -2, -4, -10, 4, 0, 3, 7, 0], [-9, -2, 3, 1, -9, 1, -2, -4, 0, 3, 1, -6], [6, -1, -2, -2, 0, 1, 11, -2, 3, -4, -3, 5], [0, -6, -7, 3, -5, 6, 4, 2, -2, -9, -2, 0], [-7, 2, -5, 2, -4, -1, 2, 4, -4, -4, -5, 2], [-16, 6, 0, 0, 2, 7, -8, 6, 1, 3, 0, -5], [-2, 12, 2, 2, 9, 0, -6, -2, 9, 11, 11, 3], [9, -5, 5, -3, -3, -7, 12, -7, -4, -3, 4, -1], [-2, 14, 0, 6, 1, 11, 4, 2, -1, -7, 0, 8], [3, -3, 9, -6, 0, 1, 10, -4, -6, -7, -5, -8], [-2, -11, 5, -4, -9, 7, 9, -4, 2, -13, -5, -3], [3, -8, 6, -3, 4, -8, 3, -6, -4, 5, -9, -4], [-17, 5, -6, 2, -6, 3, -3, 3, -4, -2, -8, -1], [0, -9, 2, -1, -4, -6, -2, -3, -4, 9, 1, -9], [3, 4, -5, 1, 3, 6, -5, 3, 3, -7, 0, 0], [15, -10, -1, -1, 2, -15, 2, -9, -3, 11, -2, 3], [15, -5, -1, -6, -1, -3, 7, -4, -3, 2, 6, -5], [7, -1, 10, -4, 7, 11, 6, 1, 7, -5, -2, 2], [-8, -1, 7, -8, 1, -5, 3, -4, -2, -7, -3, -1], [2, 1, -6, 2, -3, 3, -8, 12, -2, 5, 0, 3], [2, -9, 3, -4, -7, 3, 0, -3, 3, -7, -2, -4], [2, 2, -3, 4, -7, 5, 6, 2, 3, -5, 1, 1], [5, 0, -3, -1, 2, 3, -5, -3, -1, 9, 1, -2], [-8, 9, 5, 2, 1, 5, 3, 5, 0, -2, 5, 3], [-6, 1, -1, -2, -6, 3, 9, -8, -13, -10, -3, 3], [-11, -2, -10, 6, -12, 0, -10, 4, -1, -4, 3, -5], [-11, 4, -5, -1, 5, -2, 3, 2, -1, -7, 1, 1], [-5, 4, -2, 3, 5, 7, -1, 1, 4, 5, 0, 12], [9, 0, -4, -3, 0, 9, 3, 6, 4, 0, 3, 1], [9, -10, 12, -6, 5, -5, 9, -4, -2, 5, -5, -6], [0, 7, -7, 0, -2, 5, -4, 2, 1, 1, 2, 6], [11, 7, -2, 1, 17, 3, -1, -2, 5, 9, 4, 5], 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