
# q-expansion of newform 1089.3.b.f, downloaded from the LMFDB on 25 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 = 1089
weight = 3
poly_data = [121, 0, -80, 0, 71, 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], [[70, 0, -93, 0, 19, 0, -2, 0], 73], [[-371, 0, -40, 0, 38, 0, -4, 0], 73], [[0, -1623, 0, 714, 0, -160, 0, 13], 803], [[782, 0, -1016, 0, 206, 0, -14, 0], 73], [[0, 8248, 0, -3535, 0, 493, 0, -25], 803], [[0, -51, 0, 2142, 0, -480, 0, 39], 803], [[0, 118, 0, -2547, 0, 591, 0, -43], 803]]

hecke_ring_character_values = [[848, [-1, 0, 0, 0, 0, 0, 0, 0]], [244, [1, 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, -1, -1], [0, 0, 0, 1, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0], [0, -4, 0, 0, -1, 0, 0, 0], [0, 0, 0, -5, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 5], [0, -10, 0, 0, 0, 0, 0, 0], [-25, 0, -1, 0, 0, 0, 0, 0], [-15, 0, -5, 0, 0, 0, 0, 0], [0, 10, 0, 0, -5, 0, 0, 0], [0, 0, 0, 5, 0, -6, 0, 0], [0, 0, 0, 0, 0, 0, -10, 5], [0, 0, 0, 0, 0, 0, 5, -15], [0, 0, 0, 0, 0, 0, -2, 0], [0, 0, 0, -20, 0, 5, 0, 0], [20, 0, -10, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 8, 25], [0, 0, 0, 14, 0, 4, 0, 0], [0, 0, 0, -15, 0, 5, 0, 0], [0, -16, 0, 0, -5, 0, 0, 0], [0, 0, 0, 0, 0, 0, -15, 10], [-55, 0, 15, 0, 0, 0, 0, 0], [0, -10, 0, 0, 5, 0, 0, 0], [-15, 0, 5, 0, 0, 0, 0, 0], [0, -26, 0, 0, 10, 0, 0, 0], [0, 0, 0, -30, 0, 5, 0, 0], [0, 0, 0, 0, 0, 0, -35, 0], [0, 0, 0, -15, 0, -11, 0, 0], [0, 10, 0, 0, -5, 0, 0, 0], [0, 0, 0, 0, 0, 0, 5, 10], [0, 0, 0, 15, 0, -20, 0, 0], [0, -10, 0, 0, 15, 0, 0, 0], [0, 0, 0, 45, 0, 15, 0, 0], [30, 0, -20, 0, 0, 0, 0, 0], [-150, 0, -10, 0, 0, 0, 0, 0], [0, -14, 0, 0, -15, 0, 0, 0], [0, -54, 0, 0, -6, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 38], [-175, 0, -21, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 40, 15], [0, 0, 0, 10, 0, -4, 0, 0], [0, -74, 0, 0, 9, 0, 0, 0], [20, 0, 50, 0, 0, 0, 0, 0], [0, 0, 0, -65, 0, -20, 0, 0], [200, 0, 30, 0, 0, 0, 0, 0], [0, 46, 0, 0, -6, 0, 0, 0], [190, 0, 0, 0, 0, 0, 0, 0], [0, -74, 0, 0, -5, 0, 0, 0], [0, 10, 0, 0, 25, 0, 0, 0], [0, 0, 0, 40, 0, 30, 0, 0], [0, 0, 0, 0, 0, 0, -18, 0], [0, 0, 0, 0, 0, 0, 45, -40], [0, -74, 0, 0, -11, 0, 0, 0], [0, 0, 0, 0, 0, 0, 37, -25], [0, 0, 0, 45, 0, 15, 0, 0], [0, 0, 0, 36, 0, 31, 0, 0], [0, -100, 0, 0, -15, 0, 0, 0], [0, 0, 0, -81, 0, 14, 0, 0], [0, -46, 0, 0, 30, 0, 0, 0], [0, 0, 0, 1, 0, -34, 0, 0], [0, 0, 0, 0, 0, 0, -52, -75], [-330, 0, 10, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 45, 75], [-300, 0, 44, 0, 0, 0, 0, 0], [0, 0, 0, -20, 0, 4, 0, 0], [0, 4, 0, 0, -9, 0, 0, 0], [0, 0, 0, 60, 0, 5, 0, 0], [0, 0, 0, 0, 0, 0, -15, -110], [0, -120, 0, 0, -30, 0, 0, 0], [170, 0, 60, 0, 0, 0, 0, 0], [0, 0, 0, 36, 0, -49, 0, 0], [-30, 0, 50, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -20, 45], [0, 0, 0, 0, 0, 0, 127, 25], [-345, 0, 5, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 13, -72], [0, 0, 0, -30, 0, 40, 0, 0], [0, 0, 0, 0, 0, 0, 128, -32], [-275, 0, -9, 0, 0, 0, 0, 0], [0, -10, 0, 0, -15, 0, 0, 0], [-260, 0, -80, 0, 0, 0, 0, 0], [0, 0, 0, -15, 0, 15, 0, 0], [0, 0, 0, 0, 0, 0, -20, 90], [0, 0, 0, 0, 0, 0, 55, 30], [0, 0, 0, -24, 0, -44, 0, 0], [0, 150, 0, 0, -60, 0, 0, 0], [-5, 0, -5, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -10, 40], [0, 50, 0, 0, -15, 0, 0, 0], [-60, 0, -50, 0, 0, 0, 0, 0], [0, 150, 0, 0, 0, 0, 0, 0], [-548, 0, 0, 0, 0, 0, 0, 0], [0, 66, 0, 0, 9, 0, 0, 0], [0, 0, 0, 0, 0, 0, -87, -75], [0, 0, 0, 0, 0, 0, 127, 12], [0, 0, 0, 89, 0, -56, 0, 0], [0, 0, 0, -30, 0, -65, 0, 0], [0, 0, 0, 171, 0, 6, 0, 0], [0, -114, 0, 0, 30, 0, 0, 0], [0, 104, 0, 0, -10, 0, 0, 0], [0, -100, 0, 0, 15, 0, 0, 0], [0, 0, 0, 75, 0, 10, 0, 0], [-395, 0, 15, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -100, -80], [0, -124, 0, 0, 15, 0, 0, 0], [0, 0, 0, 0, 0, 0, -138, -3], [0, 0, 0, -90, 0, -110, 0, 0], [0, 0, 0, 85, 0, -29, 0, 0], [0, 0, 0, 80, 0, 31, 0, 0], [0, 0, 0, 0, 0, 0, -5, 190], [200, 0, -6, 0, 0, 0, 0, 0], [765, 0, 25, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -33, -48], [460, 0, -110, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 90, -45], [0, 0, 0, 0, 0, 0, -25, -65], [0, -220, 0, 0, 0, 0, 0, 0], [780, 0, -50, 0, 0, 0, 0, 0], [0, 0, 0, 214, 0, -66, 0, 0], [0, 106, 0, 0, 45, 0, 0, 0], [0, 0, 0, 0, 0, 0, -130, -100], [-482, 0, 100, 0, 0, 0, 0, 0], [0, 110, 0, 0, 50, 0, 0, 0], [-100, 0, 14, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 20, 195], [-410, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 106, 0, 1, 0, 0], [0, 0, 0, 95, 0, 20, 0, 0], [0, -84, 0, 0, 54, 0, 0, 0], [27, 0, -125, 0, 0, 0, 0, 0], [-655, 0, 95, 0, 0, 0, 0, 0], [0, 280, 0, 0, -75, 0, 0, 0], [0, 0, 0, 100, 0, 120, 0, 0], [0, 0, 0, 0, 0, 0, 75, 245], [0, 0, 0, 255, 0, 6, 0, 0], [0, 0, 0, 0, 0, 0, 95, 165], [0, -30, 0, 0, -75, 0, 0, 0], [0, 0, 0, -125, 0, 0, 0, 0], [0, -110, 0, 0, -5, 0, 0, 0], [1110, 0, -40, 0, 0, 0, 0, 0], [0, -184, 0, 0, 69, 0, 0, 0], [500, 0, 6, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -108, 25], [0, 0, 0, 154, 0, 119, 0, 0], [0, -54, 0, 0, 45, 0, 0, 0], [-300, 0, -196, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -150, -215], [0, 0, 0, 140, 0, 161, 0, 0], [0, 0, 0, 0, 0, 0, -137, 0], [1050, 0, 100, 0, 0, 0, 0, 0], [0, -156, 0, 0, 36, 0, 0, 0], [370, 0, -140, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 268, 183], [0, 0, 0, -95, 0, -25, 0, 0], [0, 0, 0, 0, 0, 0, 37, -150], [0, 0, 0, -30, 0, -56, 0, 0], [0, -290, 0, 0, 85, 0, 0, 0], [0, 0, 0, 0, 0, 0, -20, -80], [0, 46, 0, 0, -91, 0, 0, 0], [0, 0, 0, 75, 0, -9, 0, 0], [0, 0, 0, 0, 0, 0, 48, -162], [0, 0, 0, 0, 0, 0, -45, -30], [0, 0, 0, 0, 0, 0, -280, 105], [-425, 0, 171, 0, 0, 0, 0, 0], [0, 0, 0, -210, 0, 21, 0, 0], [0, 0, 0, 310, 0, -100, 0, 0], [0, 0, 0, 0, 0, 0, 15, 315], [0, 310, 0, 0, -35, 0, 0, 0], [425, 0, -129, 0, 0, 0, 0, 0], [0, -380, 0, 0, -90, 0, 0, 0], [0, 0, 0, 140, 0, -126, 0, 0], [50, 0, -204, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 37, 50], [0, 0, 0, -125, 0, 130, 0, 0], [0, 0, 0, 0, 0, 0, 93, 75], [0, 0, 0, 225, 0, -49, 0, 0], [0, 0, 0, -290, 0, -5, 0, 0], [365, 0, 125, 0, 0, 0, 0, 0], [0, 350, 0, 0, -55, 0, 0, 0], [-285, 0, 25, 0, 0, 0, 0, 0], [0, 26, 0, 0, -15, 0, 0, 0], [0, 0, 0, 0, 0, 0, -170, -15], [0, 0, 0, 0, 0, 0, -205, -105], [0, 0, 0, 94, 0, 59, 0, 0], [-1330, 0, -60, 0, 0, 0, 0, 0], [0, 0, 0, 20, 0, 90, 0, 0], [0, -130, 0, 0, -15, 0, 0, 0], [45, 0, -155, 0, 0, 0, 0, 0], [0, -356, 0, 0, 61, 0, 0, 0], [8, 0, -100, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 303, 75], [0, -436, 0, 0, -70, 0, 0, 0], [0, 0, 0, 0, 0, 0, 355, -140], [0, 0, 0, -10, 0, 30, 0, 0], [-565, 0, 165, 0, 0, 0, 0, 0], [0, 474, 0, 0, 51, 0, 0, 0], [0, 184, 0, 0, 90, 0, 0, 0], [0, -50, 0, 0, 95, 0, 0, 0], [0, 0, 0, -185, 0, -95, 0, 0], [-765, 0, -55, 0, 0, 0, 0, 0], [0, 0, 0, 340, 0, -160, 0, 0], [0, 0, 0, 0, 0, 0, -440, 60], [0, 0, 0, 0, 0, 0, -185, -215], [-1017, 0, -25, 0, 0, 0, 0, 0], [0, 114, 0, 0, 51, 0, 0, 0], [0, -540, 0, 0, -45, 0, 0, 0], [0, 0, 154, 0, 0, 0, 0, 0], [0, 0, 0, -46, 0, 164, 0, 0], [0, 0, 0, 0, 0, 0, 375, -175], [585, 0, 265, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -210, 210], [0, -640, 0, 0, -100, 0, 0, 0], [465, 0, 25, 0, 0, 0, 0, 0], [0, 0, 0, 539, 0, -91, 0, 0], [0, -60, 0, 0, -75, 0, 0, 0], [0, 0, 0, 0, 0, 0, 140, -245], [0, 0, 0, 0, 0, 0, -155, 255], [0, 0, 0, -160, 0, -55, 0, 0], [0, 0, 0, -145, 0, -95, 0, 0], [0, 0, 0, 0, 0, 0, -373, -50], [2070, 0, 20, 0, 0, 0, 0, 0], [0, -284, 0, 0, 94, 0, 0, 0], [0, 0, 0, -110, 0, 205, 0, 0], [0, 0, 0, 0, 0, 0, 105, 130], [0, 0, 0, 0, 0, 0, 88, 125], [0, 0, 0, 321, 0, -59, 0, 0], [0, 170, 0, 0, 80, 0, 0, 0], [950, 0, -60, 0, 0, 0, 0, 0], [0, 0, 0, 15, 0, 110, 0, 0], [0, 0, 0, -255, 0, -125, 0, 0], [0, -80, 0, 0, 25, 0, 0, 0], [140, 0, -280, 0, 0, 0, 0, 0], [0, -484, 0, 0, -136, 0, 0, 0], [-150, 0, 34, 0, 0, 0, 0, 0], [0, 386, 0, 0, -70, 0, 0, 0], [0, 0, 0, 0, 0, 0, 12, 0], [0, 0, 0, 0, 0, 0, -290, 235], [0, 0, 0, 0, 0, 0, -350, -200], [0, 0, 0, -75, 0, 30, 0, 0], [35, 0, -385, 0, 0, 0, 0, 0], [-350, 0, -84, 0, 0, 0, 0, 0], [0, 464, 0, 0, 125, 0, 0, 0], [0, -300, 0, 0, 120, 0, 0, 0], [610, 0, -180, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -318, -300], [0, 0, 0, -195, 0, 190, 0, 0], [0, -704, 0, 0, 0, 0, 0, 0], [0, 0, 0, 304, 0, 119, 0, 0], [0, -430, 0, 0, 45, 0, 0, 0], [0, 0, 0, 0, 0, 0, 220, -135], [-1095, 0, -125, 0, 0, 0, 0, 0], [0, 394, 0, 0, 125, 0, 0, 0], [0, -280, 0, 0, -175, 0, 0, 0], [-600, 0, -46, 0, 0, 0, 0, 0], [0, 0, 0, -111, 0, 4, 0, 0], [0, 0, 0, 0, 0, 0, -135, -255], [0, 0, 0, 310, 0, 46, 0, 0], [0, 0, 0, -309, 0, -169, 0, 0], [0, -354, 0, 0, -75, 0, 0, 0], [0, 0, 0, 150, 0, 255, 0, 0], [0, 0, 0, -124, 0, -69, 0, 0], [0, 0, 0, 0, 0, 0, 115, -190], [1900, 0, -150, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -33, -223], [0, 0, 0, 0, 0, 0, -217, 450], [0, 0, 0, -405, 0, -226, 0, 0], [0, -494, 0, 0, -110, 0, 0, 0], [-455, 0, 175, 0, 0, 0, 0, 0], [290, 0, 0, 0, 0, 0, 0, 0], [785, 0, 65, 0, 0, 0, 0, 0], [0, 0, 0, -315, 0, -315, 0, 0], [0, 0, 0, -486, 0, -86, 0, 0], [1390, 0, 120, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -540, -170], [0, 0, 0, -30, 0, 185, 0, 0], [0, 0, 0, -300, 0, 50, 0, 0], [0, 160, 0, 0, -50, 0, 0, 0], [0, -1026, 0, 0, -75, 0, 0, 0], [100, 0, -234, 0, 0, 0, 0, 0], [0, 836, 0, 0, 50, 0, 0, 0], [0, 0, 0, -270, 0, -285, 0, 0], [0, 0, 0, 45, 0, 54, 0, 0], [0, 0, 0, 0, 0, 0, 302, 387], [55, 0, -395, 0, 0, 0, 0, 0], [0, 586, 0, 0, -6, 0, 0, 0], [-1700, 0, 174, 0, 0, 0, 0, 0], [0, 100, 0, 0, 5, 0, 0, 0], [0, 0, 0, 0, 0, 0, 3, -625], [0, 0, 0, 0, 0, 0, 140, 180], [0, 636, 0, 0, -111, 0, 0, 0], [0, -180, 0, 0, 90, 0, 0, 0], [0, 0, 0, 194, 0, -131, 0, 0], [0, -190, 0, 0, -55, 0, 0, 0], [1400, 0, 350, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 195, 745], [0, -60, 0, 0, -105, 0, 0, 0], [0, 0, 0, 575, 0, 224, 0, 0], [0, 0, 0, 124, 0, 84, 0, 0], [0, 834, 0, 0, -84, 0, 0, 0], [0, 0, 0, 425, 0, -135, 0, 0]]
