
# q-expansion of newform 1152.3.m.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 = 1152
weight = 3
poly_data = [65536, 0, -24576, -4096, 2560, 3584, 1408, -512, -496, -128, 88, 56, 10, -4, -6, 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], [[-753664, -528384, -23552, 112128, 73472, 1792, -25632, -15600, -2928, 1848, 1156, 170, -168, -134, -18, 5], 12288], [[10231808, 3719168, -3177472, -2271744, -499456, 454272, 450272, 88528, -66544, -59096, -11868, 5530, 4536, 1234, -626, -347], 368640], [[-1228800, -172032, 537600, 350720, 73472, -60800, -52256, -5136, 11728, 8376, 1684, -610, -528, -90, 110, 47], 40960], [[1974272, 837632, -429568, -377856, -110464, 67008, 75248, 19792, -8056, -9224, -2082, 730, 714, 226, -89, -53], 30720], [[41107456, 22466560, -4517888, -7073280, -3026432, 829440, 1598464, 641360, -19520, -156376, -54888, 5306, 14076, 6626, -448, -751], 368640], [[14581760, 13664256, 4019200, -1328640, -1689856, -394880, 442848, 431088, 157776, -2088, -25772, -9010, 2064, 3430, 1230, 119], 122880], [[-12972032, -9620480, -88064, 2280960, 1233664, -54240, -517568, -275680, -35600, 40172, 22056, 998, -3852, -2572, -244, 197], 92160], [[-26214400, -4268032, 12336128, 6971904, 775424, -1748352, -1269472, -106640, 278960, 189400, 26652, -21122, -14184, -2666, 2242, 1063], 368640], [[-8568832, -20942848, -16144384, -2188800, 2557952, 1824000, 66368, -716144, -468448, -122744, 28992, 30658, 5076, -4022, -3884, -995], 368640], [[-6232064, -3969536, 538624, 1184256, 502912, -102288, -257408, -113704, -1448, 25118, 9756, -622, -2232, -1153, 20, 134], 46080], [[-6053888, 3770368, 7066624, 2511360, -303872, -897600, -372128, 127664, 190768, 82544, 1188, -12838, -5256, 302, 1574, 545], 184320], [[13975552, -11743232, -18839552, -6455808, 922624, 2421504, 933184, -401488, -508256, -219304, -528, 35366, 13356, -1186, -4300, -1417], 368640], [[-5570560, -1472512, 2561024, 1423872, 217088, -338496, -271264, -23888, 59984, 41008, 6228, -4406, -2856, -578, 526, 265], 61440], [[-25296896, -3891200, 12133888, 6497280, 602752, -1754880, -1210544, -52720, 302680, 190016, 22218, -22786, -13986, -2326, 2693, 1181], 184320], [[80101376, 28983296, -21873664, -16501248, -3795712, 3329664, 3347744, 698896, -462928, -427352, -85236, 37810, 33552, 9178, -4382, -2519], 368640]]

hecke_ring_character_values = [[127, [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], [901, [0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], [641, [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]]
aps_data = [[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, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 0, 2, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, -1, 1, 0, -1, 0, 1, 0, 0, 0, -1, -1], [0, 1, 0, 1, 0, 0, -1, -1, 0, -1, 0, 0, 0, 0, 0, 0], [2, 0, -2, 0, 0, 0, -1, 0, 0, 1, 0, 0, -1, 2, 0, 1], [8, -1, 0, -2, 0, 2, 0, 0, 0, 0, -2, 2, 0, -2, 2, 0], [2, 0, 2, -1, 0, 3, -1, -2, 1, 0, 0, 0, 0, 0, 2, 1], [0, 0, 8, 0, 0, 2, 0, -1, -1, 1, 1, -2, -1, 0, 0, 2], [6, 0, -6, -1, 0, 0, 0, 0, -1, 2, 0, -1, -2, -1, 0, -1], [0, 0, 0, 0, -3, -2, -2, 1, 3, -1, -2, 2, 0, 2, 2, 1], [-10, 0, -10, 2, 0, 2, -1, -1, -2, 0, 1, 0, 0, 0, -2, 1], [0, 0, 24, 0, 3, -2, 2, 0, -2, 0, 2, 2, 0, -2, -2, -2], [-10, 2, 10, 3, 2, 0, 1, 0, 3, 2, 0, 1, 2, -2, 0, -1], [-8, 0, -8, -2, 0, -4, -2, 2, 2, 0, -2, 0, 0, 0, 0, 2], [2, -4, 2, 1, 4, -1, 1, 2, -1, 0, 5, 0, 0, 0, 1, -1], [-20, 1, 20, -2, 1, 0, -2, 0, -2, -2, 0, -2, -4, 0, 0, -2], [-32, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 6, 0, 4, -3, -1, -1, 0, 1, 0, 3, -1, 1, 1, -4], [0, 0, -8, 0, 6, -6, 0, 0, -5, 0, 2, 6, -2, 0, 0, 0], [-10, 0, 10, 0, 0, 0, -2, 0, 0, -2, 0, 3, -3, 4, 0, 2], [0, 0, -10, 0, 6, 0, 0, 0, -8, 0, 2, 0, -2, 0, 0, -4], [0, 6, 0, 2, 0, 2, -4, -2, 0, -2, 6, 2, 4, 2, -2, 0], [0, -10, 0, -6, -10, 0, 0, 0, -6, 0, 0, -1, -4, -2, 0, -2], [24, 0, 0, -5, 0, -4, -6, -1, 0, -1, -5, -4, -5, 0, 0, 0], [0, 0, 0, 6, 0, 6, 0, 0, -6, 0, 6, 0, 0, 0, -4, 0], [32, 8, 32, 5, -8, -3, -3, -4, -5, 0, -1, 0, 0, 0, 5, 3], [14, 8, 0, 12, 0, -6, 2, 2, 0, 2, 0, -6, -2, 2, -2, 0], [0, 0, 32, 0, 10, 2, 0, 3, -5, -3, -3, -2, 3, 0, 0, -2], [-20, 1, 20, 4, 1, 0, 4, 0, 4, 8, 0, -10, -2, 0, 0, 4], [0, 0, 0, 0, -15, -2, 10, -1, -3, 1, 12, 2, -2, -10, -10, -13], [-12, -5, -12, -4, 5, 18, -4, 4, 4, 0, -10, 0, 0, 0, 8, 4], [52, 10, -52, 2, 10, 0, -6, 0, 2, -4, 0, 1, -8, 4, 0, -2], [16, 4, 0, -3, 0, 8, -6, -3, 0, -3, -11, 8, -7, -4, 4, 0], [-4, -12, -4, -1, 12, -5, -1, -8, 1, 0, 3, 0, 0, 0, -3, 1], [-30, 0, 30, -8, 0, 0, 1, 0, -8, 7, 0, -10, -1, -2, 0, -1], [-56, -7, 0, -4, 0, -8, -4, 0, 0, 0, -4, -8, -6, 2, -2, 0], [-42, -12, -42, 7, 12, -7, -1, -6, -7, 0, -2, 0, 0, 0, 0, 1], [36, 0, -36, -2, 0, 0, 2, 0, -2, 10, 0, 14, 0, -8, 0, -6], [42, -16, -42, 1, -16, 0, -6, 0, 1, 6, 0, -7, -6, 7, 0, 1], [0, 0, -56, 0, -15, -2, 6, 6, -4, -6, 0, 2, 6, -6, -6, -10], [-4, 24, 0, 0, 0, -7, 5, 5, 0, 5, -10, -7, -1, -9, 9, 0], [26, -14, -26, 3, -14, 0, -7, 0, 3, -2, 0, -9, 6, 6, 0, -1], [-64, -14, 0, -3, 0, -8, -2, 3, 0, 3, 3, -8, 3, 0, 0, 0], [-12, 7, 12, 4, 7, 0, -4, 0, 4, 12, 0, -6, -2, 8, 0, 4], [0, 0, -120, 0, -20, 22, -4, 2, 3, -2, 8, -22, -12, 4, 4, 0], [-78, 0, 78, -12, 0, 0, 6, 0, -12, -6, 0, 5, 3, -4, 0, 2], [50, 12, -50, -11, 12, 0, 6, 0, -11, 2, 0, 13, 6, -1, 0, 5], [0, 0, 26, 0, -26, 14, -14, -6, 24, 6, -14, -14, 0, 14, 14, 20], [0, 0, 56, 0, 15, -4, 2, 8, 12, -8, -8, 4, 10, -2, -2, 2], [-28, -30, 0, 6, 0, -15, 7, -1, 0, -1, 10, -15, 3, 7, -7, 0], [114, 0, 114, -8, 0, -33, 12, 16, 8, 0, 15, 0, 0, 0, -8, -12], [-10, -30, 0, -12, 0, 16, -12, -4, 0, -4, -6, 16, -2, -4, 4, 0], [-48, 30, 0, -12, 0, -6, -4, 0, 0, 0, 2, -6, 6, -4, 4, 0], [56, -16, 56, -16, 16, 13, 0, 0, 16, 0, -2, 0, 0, 0, 2, 0], [0, 0, 72, 0, 12, 10, -12, -4, 17, 4, -10, -10, -2, 12, 12, 12], [-54, -12, 54, -9, -12, 0, 6, 0, -9, 2, 0, -33, -10, 1, 0, 7], [0, 0, 50, 0, 34, 6, -2, 2, 16, -2, -14, -6, 12, 2, 2, 4], [-40, 9, -40, 10, -9, -12, 10, 18, -10, 0, 6, 0, 0, 0, 8, -10], [-78, -18, 78, 1, -18, 0, 3, 0, 1, 6, 0, 19, 6, 2, 0, 5], [20, -9, -20, 14, -9, 0, 2, 0, 14, -6, 0, -2, 16, 8, 0, 10], [64, 6, 0, -16, 0, 6, 20, 4, 0, 4, 2, 6, 6, -4, 4, 0], [0, 0, -44, 0, -28, -33, 9, 3, 0, -3, -10, 33, 19, -9, -9, 4], [24, 14, 24, -2, -14, 29, -14, -8, 2, 0, -22, 0, 0, 0, 26, 14], [-4, 15, -4, -12, -15, -10, 0, -8, 12, 0, 14, 0, 0, 0, -8, 0], [-90, -8, 0, -36, 0, 25, 5, 11, 0, 11, -4, 25, -7, 3, -3, 0], [142, 12, 142, 12, -12, -1, 2, -6, -12, 0, -7, 0, 0, 0, -12, -2], [-62, -16, -62, 5, 16, 17, -11, 6, -5, 0, -25, 0, 0, 0, 19, 11], [34, -10, 0, 24, 0, -22, -2, 6, 0, 6, 6, -22, -16, 22, -22, 0], [-64, -62, 0, 14, 0, 12, 12, 2, 0, 2, 6, 12, 10, -4, 4, 0], [0, 0, 152, 0, -18, 18, -4, 1, 9, -1, -5, -18, 1, 4, 4, -22], [52, -20, -52, 3, -20, 0, 28, 0, 3, -16, 0, 27, 24, -25, 0, 3], [46, 0, 46, -30, 0, 16, 1, 1, 30, 0, -15, 0, 0, 0, 10, -1], [0, 0, -72, 0, -41, 14, -6, -20, 12, 20, -18, -14, 12, 6, 6, 30], [66, -16, -66, 5, -16, 0, 21, 0, 5, -2, 0, 9, -14, -16, 0, 5], [-132, 16, -132, -25, -16, 53, -17, -16, 25, 0, -27, 0, 0, 0, 27, 17], [0, 19, 0, -45, 0, 16, -19, -7, 0, -7, -28, 16, -28, 0, 0, 0], [0, 0, -20, 0, -6, 56, -28, 8, 46, -8, -26, -56, -2, 28, 28, 26], [-178, 0, 178, -28, 0, 0, 2, 0, -28, 10, 0, -17, 5, 20, 0, 22], [18, 32, -18, 23, 32, 0, 0, 0, 23, 18, 0, -27, -2, 15, 0, 15], [0, 0, 200, 0, 15, 20, 2, -18, -26, 18, 22, -20, -20, -2, -2, -18], [28, 0, 0, -16, 0, -40, -4, -10, 0, -10, -10, -40, 2, -12, 12, 0], [112, -6, 0, -41, 0, -4, 10, 9, 0, 9, -19, -4, 13, -32, 32, 0], [134, 0, 134, -36, 0, 51, -14, -6, 36, 0, -19, 0, 0, 0, 20, 14], [16, -3, 0, 15, 0, -32, 25, 9, 0, 9, 2, -32, 6, -4, 4, 0], [0, 0, 48, 0, -2, 10, 18, -10, 14, 10, -6, -10, 24, -18, -18, 10], [70, -14, 70, -1, 14, -37, 11, 10, 1, 0, -10, 0, 0, 0, -12, -11], [0, 0, 144, 0, 18, -10, 32, 9, 1, -9, 23, 10, 9, -32, -32, -30], [230, -28, -230, 8, -28, 0, -22, 0, 8, 14, 0, 3, -11, 28, 0, 6], [0, 0, -256, 0, 58, 34, 4, -8, 30, 8, 2, -34, 2, -4, -4, 24], [-8, 78, 0, -17, 0, 24, -4, 0, 0, 0, -16, 24, 0, -16, 16, 0], [-216, 11, -216, 12, -11, -2, 18, 14, -12, 0, 12, 0, 0, 0, -28, -18], [48, 27, -48, 14, 27, 0, -10, 0, 14, 6, 0, -96, -18, 24, 0, 14], [256, 6, 0, 46, 0, -28, -28, -10, 0, -10, 18, -28, 22, -4, 4, 0], [126, -18, 126, 23, 18, 1, -5, -6, -23, 0, -4, 0, 0, 0, -26, 5], [0, 0, 0, 0, -15, 16, 4, 11, 7, -11, -10, -16, 14, -4, -4, -27], [-182, -10, -182, 16, 10, -80, 29, 5, -16, 0, 21, 0, 0, 0, -26, -29], [-118, 8, -118, 13, -8, 7, 21, 18, -13, 0, 43, 0, 0, 0, -29, -21], [-146, 0, 146, -10, 0, 0, 1, 0, -10, 3, 0, 70, 23, 18, 0, 19], [-24, 22, -24, -18, -22, -33, 18, 8, 18, 0, -20, 0, 0, 0, 24, -18], [54, 36, -54, 20, 36, 0, 4, 0, 20, -16, 0, -15, 17, 16, 0, 20], [0, 0, 176, 0, 11, -22, -22, -5, 1, 5, -18, 22, -4, 22, 22, 19], [96, 13, 96, -18, -13, 68, -18, 14, 18, 0, -14, 0, 0, 0, 32, 18], [-2, -24, 0, 4, 0, -1, -5, 9, 0, 9, 32, -1, 19, 13, -13, 0], [-140, -25, -140, -20, 25, -56, 6, 6, 20, 0, 14, 0, 0, 0, -20, -6], [46, -8, 0, 44, 0, 6, 46, 14, 0, 14, 40, 6, 34, 6, -6, 0], [424, -7, 0, -62, 0, -6, -12, -6, 0, -6, -8, -6, -26, 18, -18, 0], [0, 0, 10, 0, -4, 45, 7, -25, -48, 25, 24, -45, -17, -7, -7, -4], [0, 0, -80, 0, -84, -42, 12, 3, 3, -3, 9, 42, 3, -12, -12, 2], [108, -16, -108, 11, -16, 0, -10, 0, 11, -40, 0, -27, -8, -19, 0, -29], [0, 0, 398, 0, 52, -42, 18, -18, -12, 18, 0, 42, 18, -18, -18, 4], [-144, -33, -144, -16, 33, 20, -28, 4, 16, 0, -8, 0, 0, 0, 8, 28], [-152, -8, 0, 57, 0, -28, -20, -24, 0, -24, 12, -28, -8, 20, 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