
# q-expansion of newform 405.4.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 = 405
weight = 4
poly_data = [614656, 0, 4069504, 0, 6555196, 0, 2881141, 0, 571975, 0, 59128, 0, 3268, 0, 91, 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], [[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 1], [[11, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 1], [[0, 19, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 1], [[15782656, 0, 22574480, 0, 1226376, 0, -1713581, 0, -370126, 0, -29466, 0, -1022, 0, -13, 0], 176256], [[17199616, 0, 85224032, 0, 36266220, 0, 5621959, 0, 387218, 0, 11574, 0, 94, 0, -1, 0], 352512], [[-13692160, 0, -30262848, 0, 8366676, 0, 5902643, 0, 968298, 0, 68574, 0, 2222, 0, 27, 0], 117504], [[-8294528, 0, -64228384, 0, -73691748, 0, -22981937, 0, -3084622, 0, -200394, 0, -6194, 0, -73, 0], 176256], [[-833385728, -864930048, -2430051904, -3094154400, -1674133020, -1749552876, -443406929, -399823161, -55246030, -45044382, -3454794, -2626650, -104762, -75642, -1225, -849], 34546176], [[833385728, -864930048, 2430051904, -3094154400, 1674133020, -1749552876, 443406929, -399823161, 55246030, -45044382, 3454794, -2626650, 104762, -75642, 1225, -849], 34546176], [[-11354880, -17007872, -52858752, -49592896, -50432796, -34165980, -14215017, -9049121, -1750686, -1127470, -105210, -70506, -3018, -2138, -33, -25], 705024], [[0, -100002400, 0, -13608404, 0, -63425913, 0, -25802659, 0, -3850424, 0, -263820, 0, -8407, 0, -101], 2878848], [[-11354880, 17007872, -52858752, 49592896, -50432796, 34165980, -14215017, 9049121, -1750686, 1127470, -105210, 70506, -3018, 2138, -33, 25], 705024], [[0, 79369792, 0, -1116316120, 0, -1022927202, 0, -295408283, 0, -38436238, 0, -2465730, 0, -75908, 0, -895], 17273088], [[0, -640762624, 0, -1314355088, 0, -689377056, 0, -157923085, 0, -18206894, 0, -1088058, 0, -31990, 0, -365], 5757696], [[0, -475010752, 0, -1030303640, 0, -455445594, 0, -87674005, 0, -8743514, 0, -468198, 0, -12712, 0, -137], 1016064]]

hecke_ring_character_values = [[326, [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], [82, [-1, 0, 0, 0, 0, 0, 0, 0, 0, 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, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [5, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 1], [0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 1, 1, -1, 0, 0, -1], [0, 0, -1, 0, 1, 0, -1, -2, -1, 1, 1, 0, 1, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0, -1, -1, 2, 1, -2, -3, -1, 1], [-28, 0, 7, 0, 1, 1, -1, 1, 0, 0, 2, 0, 2, 0, 0, 0], [1, 0, 0, 0, -2, -5, -2, -1, -4, 4, -2, 0, -2, 0, 0, 0], [0, 23, 0, -6, 0, 0, 0, 0, 6, 6, 1, 7, -1, -1, -2, 0], [38, 0, 2, 0, -4, 1, -2, 3, -1, 1, 0, 0, 0, 0, 0, 0], [0, -35, 0, -1, 0, 0, 0, 0, 9, 9, 1, 3, -1, -2, -2, -1], [0, 0, 0, 0, 0, 0, 0, 0, -2, -2, 3, 5, -3, 6, 8, 0], [0, -1, 0, -3, 0, 0, 0, 0, 1, 1, 2, -5, -2, -12, 5, 1], [-125, 0, 27, 0, 3, -1, 3, 1, -3, 3, 2, 0, 2, 0, 0, 0], [6, 0, -3, 0, 6, 1, 2, -8, -9, 9, 5, 0, 5, 0, 0, 0], [0, 22, 0, 5, 0, 0, 0, 0, 12, 12, -1, -7, 1, 7, -18, -1], [191, 0, 2, 0, -2, 9, 8, 3, 1, -1, 3, 0, 3, 0, 0, 0], [0, -89, 0, -6, 0, 0, 0, 0, 3, 3, 5, -7, -5, 7, -7, 6], [5, 0, -8, 0, 3, -3, -9, 0, -3, 3, -6, 0, -6, 0, 0, 0], [0, -22, 0, 13, 0, 0, 0, 0, -11, -11, -2, 3, 2, 15, -19, 5], [-250, 0, 23, 0, 11, -5, -11, -15, 1, -1, 3, 0, 3, 0, 0, 0], [0, -4, 0, 25, 0, 0, 0, 0, -15, -15, 7, -22, -7, 23, 34, 1], [441, 0, -8, 0, 5, 1, 1, 14, -1, 1, -5, 0, -5, 0, 0, 0], [0, -160, 0, -15, 0, 0, 0, 0, -12, -12, -6, -8, 6, 5, 7, 9], [0, -25, 0, -11, 0, 0, 0, 0, 9, 9, 3, 34, -3, -6, -26, -15], [-57, 0, 31, 0, -3, 10, 11, -2, 24, -24, -5, 0, -5, 0, 0, 0], [0, 51, 0, -15, 0, 0, 0, 0, 10, 10, 6, 7, -6, 12, 1, 9], [0, -12, 0, 6, 0, 0, 0, 0, -39, -39, 15, 43, -15, -36, 4, -6], [749, 0, 22, 0, 19, -17, -7, -32, 23, -23, 23, 0, 23, 0, 0, 0], [0, 96, 0, -7, 0, 0, 0, 0, -37, -37, 17, 14, -17, 27, 18, 1], [-197, 0, 57, 0, -15, 23, 1, 5, 45, -45, 14, 0, 14, 0, 0, 0], [-287, 0, -43, 0, -7, 20, 19, 14, -3, 3, 25, 0, 25, 0, 0, 0], [184, 0, -24, 0, -21, 2, 1, 11, 48, -48, 17, 0, 17, 0, 0, 0], [0, -21, 0, 15, 0, 0, 0, 0, -78, -78, -22, -1, 22, -6, -1, -9], [0, -308, 0, -17, 0, 0, 0, 0, -63, -63, -26, 12, 26, -17, -19, -17], [0, 67, 0, 9, 0, 0, 0, 0, 45, 45, 18, 2, -18, 24, -7, -15], [0, -80, 0, 0, 0, 0, 0, 0, 51, 51, 21, 38, -21, -48, 14, -18], [-215, 0, -156, 0, -8, -51, 8, 7, 59, -59, -4, 0, -4, 0, 0, 0], [4, 0, 7, 0, -5, -47, -11, 11, 119, -119, -6, 0, -6, 0, 0, 0], [1258, 0, 210, 0, 1, 18, 5, -17, 64, -64, 32, 0, 32, 0, 0, 0], [0, -273, 0, -31, 0, 0, 0, 0, -105, -105, 24, -43, -24, 0, 42, -37], [0, 207, 0, -5, 0, 0, 0, 0, 41, 41, 9, -35, -9, 66, 19, 33], [203, 0, -144, 0, 42, -15, 0, -3, 81, -81, 15, 0, 15, 0, 0, 0], [66, 0, -60, 0, 3, 4, -7, -23, 96, -96, -31, 0, -31, 0, 0, 0], [0, -225, 0, 65, 0, 0, 0, 0, -81, -81, -37, -6, 37, 60, 25, 47], [0, 18, 0, -10, 0, 0, 0, 0, 104, 104, -22, -90, 22, 12, -23, 10], [460, 0, -79, 0, 15, 41, 7, 29, 90, -90, 19, 0, 19, 0, 0, 0], [0, 188, 0, -128, 0, 0, 0, 0, -93, -93, 6, -2, -6, -132, -59, -18], [-695, 0, -248, 0, 31, 83, 17, -6, 51, -51, 15, 0, 15, 0, 0, 0], [-185, 0, -4, 0, 27, 26, 19, -7, 99, -99, -2, 0, -2, 0, 0, 0], [2445, 0, 111, 0, 38, 35, 40, 54, -123, 123, -9, 0, -9, 0, 0, 0], [0, -77, 0, -48, 0, 0, 0, 0, -90, -90, -6, 167, 6, 69, -79, 18], [0, 468, 0, -111, 0, 0, 0, 0, -14, -14, -18, 124, 18, -51, -53, 33], [-1245, 0, -223, 0, 3, -14, -45, -26, 0, 0, -58, 0, -58, 0, 0, 0], [439, 0, -232, 0, -4, 147, 28, 53, -71, 71, -18, 0, -18, 0, 0, 0], [0, -472, 0, -39, 0, 0, 0, 0, -12, -12, -37, 88, 37, -73, -137, -33], [2380, 0, 143, 0, 36, -69, -24, -46, -95, 95, -56, 0, -56, 0, 0, 0], [0, 341, 0, -30, 0, 0, 0, 0, 66, 66, 3, -166, -3, -103, 20, 18], [0, 56, 0, -41, 0, 0, 0, 0, -62, -62, -41, -148, 41, -87, 145, -7], [0, -968, 0, 90, 0, 0, 0, 0, 72, 72, 14, -165, -14, 130, 129, -18], [2876, 0, 240, 0, -55, 6, -17, 5, 98, -98, -71, 0, -71, 0, 0, 0], [0, 760, 0, -60, 0, 0, 0, 0, -24, -24, 93, 32, -93, -80, 155, -12], [0, -661, 0, 82, 0, 0, 0, 0, 19, 19, -47, -89, 47, 63, -96, -16], [-958, 0, 342, 0, 10, -10, 36, 42, -85, 85, -65, 0, -65, 0, 0, 0], [0, -1012, 0, 85, 0, 0, 0, 0, -39, -39, 17, 94, -17, 263, 108, 1], [0, -129, 0, -75, 0, 0, 0, 0, -47, -47, -21, 153, 21, -198, -12, -87], [943, 0, -9, 0, 16, -44, -44, -49, 50, -50, -32, 0, -32, 0, 0, 0], [0, -645, 0, 36, 0, 0, 0, 0, -188, -188, -18, 155, 18, -75, 113, -78], [-3148, 0, 160, 0, -7, 20, 7, -17, -251, 251, 23, 0, 23, 0, 0, 0], [0, -1044, 0, 53, 0, 0, 0, 0, 174, 174, 47, -46, -47, 237, -129, 11], [0, 1011, 0, 181, 0, 0, 0, 0, -66, -66, -50, -113, 50, 180, -81, 25], [192, 0, 189, 0, 14, -22, -28, 73, -110, 110, -119, 0, -119, 0, 0, 0], [0, 340, 0, -159, 0, 0, 0, 0, 240, 240, 45, -90, -45, 297, 195, -9], [-2442, 0, -83, 0, 120, -73, -96, -124, -229, 229, 55, 0, 55, 0, 0, 0], [0, -1542, 0, 98, 0, 0, 0, 0, -42, -42, 22, -118, -22, -216, -90, 74], [3088, 0, 67, 0, -20, 2, 92, 117, 123, -123, -15, 0, -15, 0, 0, 0], [-1373, 0, -161, 0, 83, -81, -83, 65, 85, -85, -77, 0, -77, 0, 0, 0], [-3608, 0, 114, 0, 53, -86, -113, -89, 186, -186, 29, 0, 29, 0, 0, 0], [2548, 0, -720, 0, 16, -44, -116, -52, 119, -119, 40, 0, 40, 0, 0, 0], [3307, 0, -108, 0, 33, -145, -141, -44, -10, 10, -55, 0, -55, 0, 0, 0], [0, 1657, 0, 21, 0, 0, 0, 0, -174, -174, -13, 97, 13, 22, -242, 81], [679, 0, -496, 0, 45, -37, 73, -16, -207, 207, 4, 0, 4, 0, 0, 0], [0, 63, 0, 242, 0, 0, 0, 0, 227, 227, 8, 206, -8, 531, -285, 100], [-3880, 0, 513, 0, -14, 74, 104, 53, 424, -424, 4, 0, 4, 0, 0, 0], [0, -1994, 0, -106, 0, 0, 0, 0, -222, -222, 67, 74, -67, -200, 105, -58], [3042, 0, -113, 0, -77, 201, 167, 191, 146, -146, 55, 0, 55, 0, 0, 0], [0, 1733, 0, 126, 0, 0, 0, 0, 171, 171, -78, -315, 78, 5, 222, 36], [0, 781, 0, -67, 0, 0, 0, 0, 372, 372, 28, 321, -28, -156, -100, -97], [-4287, 0, 272, 0, 71, -93, 37, 106, 53, -53, -91, 0, -91, 0, 0, 0], [0, -1267, 0, -350, 0, 0, 0, 0, 222, 222, -3, 211, 3, -127, 264, -146], [2228, 0, 237, 0, 18, -50, 90, -103, -295, 295, 4, 0, 4, 0, 0, 0], [2476, 0, -21, 0, -206, -218, 46, 47, -1, 1, -134, 0, -134, 0, 0, 0], [0, 69, 0, -197, 0, 0, 0, 0, -526, -526, 57, -70, -57, 96, 59, -63], [-4421, 0, 629, 0, -236, -98, 44, -49, 156, -156, -44, 0, -44, 0, 0, 0], [1482, 0, -346, 0, 71, -69, -89, -248, -387, 387, 47, 0, 47, 0, 0, 0], [0, 659, 0, 141, 0, 0, 0, 0, 132, 132, 46, -20, -46, -274, -11, -33], [2025, 0, -999, 0, -32, -18, -22, -149, 413, -413, 17, 0, 17, 0, 0, 0], [0, -1478, 0, 74, 0, 0, 0, 0, 105, 105, -107, -353, 107, -20, 299, 8], [0, 962, 0, -56, 0, 0, 0, 0, -336, -336, -111, -32, 111, -330, -26, -162], [0, -243, 0, 96, 0, 0, 0, 0, -365, -365, -18, 100, 18, 39, 67, 186], [-1553, 0, -589, 0, 139, 103, 113, -43, 343, -343, 109, 0, 109, 0, 0, 0], [1927, 0, -293, 0, -121, 243, 31, 101, -173, 173, -92, 0, -92, 0, 0, 0], [0, -514, 0, -385, 0, 0, 0, 0, -330, -330, 116, 302, -116, 119, -162, 41], [0, -1796, 0, -222, 0, 0, 0, 0, -147, -147, 81, -5, -81, -78, 475, -12], [0, 509, 0, 180, 0, 0, 0, 0, -44, -44, 35, -7, -35, -45, -192, 190], [-3335, 0, -258, 0, 99, 71, -219, 100, -285, 285, -151, 0, -151, 0, 0, 0], [-2257, 0, 1079, 0, 13, -289, -21, -201, 431, -431, -15, 0, -15, 0, 0, 0], [0, -2424, 0, 252, 0, 0, 0, 0, 432, 432, -124, -27, 124, 186, -348, -54], [0, 1264, 0, -91, 0, 0, 0, 0, -27, -27, -219, -168, 219, -17, 97, -181], [0, -308, 0, 99, 0, 0, 0, 0, -285, -285, -75, -202, 75, 393, 176, -93], [4050, 0, 201, 0, 161, 362, -91, -74, -296, 296, 318, 0, 318, 0, 0, 0], [-413, 0, 858, 0, 63, -37, 1, 92, -432, 432, 123, 0, 123, 0, 0, 0], [-4945, 0, 430, 0, 89, 68, -155, -281, 243, -243, 110, 0, 110, 0, 0, 0], [0, -113, 0, -42, 0, 0, 0, 0, 135, 135, 298, 36, -298, 175, 219, -60], [0, -2801, 0, -409, 0, 0, 0, 0, 72, 72, 94, 278, -94, -330, -197, -25], [0, -1237, 0, -92, 0, 0, 0, 0, 301, 301, 124, 77, -124, -411, -158, -28], [-3253, 0, -84, 0, -57, 277, 117, -148, -465, 465, 151, 0, 151, 0, 0, 0], [-4354, 0, 1164, 0, -38, -400, 24, -30, 113, -113, -149, 0, -149, 0, 0, 0], [0, -419, 0, -122, 0, 0, 0, 0, -171, -171, 92, -285, -92, -227, 14, 202], [0, 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