# q-expansion of newform 80.2.j.b, downloaded from the LMFDB on 22 September 2026.

# 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 = 80
weight = 2
poly_data = [512, 0, 256, -256, -160, -224, -80, 24, 74, 70, 37, 6, -10, -14, -5, -4, 2, 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, 0, 0], 1], [[-58624, -44800, -64960, -22176, -368, 25048, 29424, 21502, 9450, -123, -4402, -4526, -2778, -797, -116, 398, 124, 129], 1280], [[6048, 4832, 7152, 2444, -44, -2972, -3408, -2471, -1033, 98, 580, 559, 331, 88, 2, -53, -19, -16], 160], [[28416, 21696, 31296, 9920, -848, -13500, -15406, -10894, -4379, 583, 2676, 2550, 1487, 375, -6, -248, -89, -75], 640], [[89600, 71296, 102976, 37376, 1200, -40528, -48500, -35846, -15784, 331, 7678, 7846, 4800, 1377, 120, -706, -258, -229], 1280], [[75264, 58816, 83776, 28832, -1872, -35116, -40294, -28628, -11749, 1284, 6810, 6492, 3793, 954, 6, -614, -217, -178], 640], [[-222976, -175616, -245696, -79200, 10928, 108280, 120736, 84014, 32794, -5723, -21346, -19710, -11242, -2685, 156, 1918, 684, 545], 1280], [[-351232, -268416, -381376, -117504, 21456, 171792, 188932, 130042, 49524, -10465, -34194, -31194, -17564, -4043, 352, 3110, 1090, 871], 1280], [[-71232, -43104, -61744, -752, 23236, 46806, 40462, 21455, 2381, -8848, -11038, -7827, -3459, -134, 672, 1013, 359, 228], 320], [[345088, 226816, 315776, 31808, -89664, -210344, -191488, -108430, -20334, 34427, 47762, 35438, 16526, 1381, -2508, -4342, -1536, -1017], 1280], [[-46208, -32320, -45504, -8592, 8624, 26316, 25600, 15672, 4196, -3435, -5762, -4568, -2280, -321, 232, 524, 186, 129], 128], [[305280, 222336, 313696, 77216, -39080, -161448, -165670, -107056, -34559, 16191, 33828, 28416, 14975, 2727, -930, -3066, -1073, -799], 640], [[-636160, -443136, -624896, -112896, 123840, 366728, 354480, 215386, 56114, -49301, -80798, -63546, -31490, -4267, 3380, 7346, 2608, 1799], 1280], [[-33024, -23168, -32672, -6240, 6024, 18760, 18318, 11252, 3063, -2417, -4104, -3268, -1639, -237, 162, 374, 133, 93], 64], [[363008, 259392, 368512, 82784, -53344, -196852, -198348, -125876, -38378, 21883, 41990, 34564, 17886, 3013, -1348, -3828, -1344, -981], 640], [[-718848, -504704, -713344, -138656, 128064, 405248, 397028, 244618, 67296, -51425, -88226, -70426, -35416, -5187, 3408, 8030, 2830, 1999], 1280], [[372864, 258560, 364800, 67376, -69792, -211368, -205144, -125222, -33350, 27653, 46182, 36566, 18238, 2547, -1844, -4198, -1484, -1039], 640], [[218944, 158976, 225296, 53272, -30132, -118616, -120714, -77418, -24294, 12619, 25220, 20982, 10958, 1929, -774, -2294, -812, -593], 320]]

hecke_ring_character_values = [[31, [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], [21, [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0]], [17, [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0]]]
aps_data = [[0, 0, 0, 0, 0, 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, -1, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [1, -1, 1, 1, -1, 1, 1, 0, 0, 0, 0, -1, 0, -1, 0, 1, 0, 0], [0, 0, -1, 0, 1, 0, -1, 0, 0, -1, 0, 0, -1, 1, 1, 0, 0, 0], [-1, 2, 0, -1, 1, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, -2, 1, 0], [1, 0, -1, 0, -1, 0, -1, 0, -1, -1, 1, 0, 0, -1, 1, 1, 0, 1], [0, 0, 1, 1, 0, 1, 0, 1, -1, 1, -2, 0, 3, 0, 1, 0, -1, -1], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 2, 0, 0, 0, 0, 0], [0, 2, 1, 0, 0, 0, 1, -1, 1, -1, 0, 0, -1, 1, -1, -2, 2, -1], [2, 0, 2, 2, -2, 2, 4, 2, -1, 0, -2, 2, 2, -2, 0, 0, -2, -1], [0, -3, -1, 0, 1, 1, 1, 0, 1, 0, -1, -1, -1, -1, 0, 0, 1, -1], [-4, 1, -1, -3, 2, -1, -3, -1, 2, 0, 1, -1, -1, 1, 0, 0, 1, 2], [-2, 0, -1, 0, -2, -1, 0, -1, -1, 1, 0, 0, -1, -2, -1, 2, 0, 1], [0, -2, -1, 0, 1, 0, 1, 0, 0, 1, 0, 0, -1, -1, 1, 0, 0, 0], [1, 2, -1, 0, -1, 0, 1, 0, 1, -1, 1, -2, -2, 1, 1, -1, 2, -1], [1, -4, 0, 2, -2, 0, 0, 2, 0, 0, -2, 1, 0, 0, -1, 0, -2, -1], [0, 1, -1, -1, 2, -1, -3, -1, 0, 0, 1, -1, -3, 1, 0, 2, 1, 0], [2, 0, 1, 0, 0, 0, -1, -1, 1, -1, 0, 0, 1, -1, 1, 0, 0, 1], [1, -2, 0, 1, -1, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, -2, -1, 0], [-4, 1, -2, -2, 4, 1, -4, 0, 1, 1, -1, 1, 0, 2, -1, -2, -1, 1], [0, 3, 3, 0, 2, -1, 3, -1, 0, 0, 1, 1, 1, 3, 0, -2, 2, 0], [2, 3, 2, 4, 0, 1, 2, 2, -1, -1, -1, 1, 0, -2, 1, 2, -1, -1], [1, -1, 2, 3, -1, 3, 2, -1, 1, 1, -2, -1, 0, -2, -1, 0, -2, -1], [-1, -1, -4, -2, -2, 1, 0, -2, -1, 1, 4, -3, -2, 0, -1, 0, 1, 1], [0, 4, -1, -1, -2, -3, -2, 1, 1, -1, 4, 2, -1, -2, -3, 2, 1, 3], [0, -4, 2, 0, 4, 0, 2, 2, 0, 2, -8, 0, 0, 2, 2, -2, -1, 0], [1, -3, 0, 0, -2, -1, 4, -4, -1, 1, 1, 0, -2, -2, 0, 2, -1, 0], [1, 1, 2, 2, 0, -1, 2, 0, 1, -1, -2, 1, -2, 0, 1, 0, 1, -1], [6, 1, 1, 1, 0, -2, 2, -1, 2, -2, 3, -1, 1, 0, 2, 0, 2, -2], [-1, -2, 3, 3, 1, 4, 1, 2, -3, 3, -7, 0, 2, 1, 3, -1, -3, -3], [-1, 1, 2, 2, 0, -1, 2, 0, 1, -1, -2, 3, 2, 0, -1, 0, 1, 1], [-1, 3, 3, 2, -5, 3, 5, 2, -4, 4, -2, 5, 6, -1, -2, 1, -5, 2], [-1, 3, 2, -4, 0, 1, -2, -4, 1, 1, 1, -2, 0, 0, 0, -2, 5, 0], [-6, -3, 1, -2, -2, 1, 1, -1, -2, 2, -1, 1, 1, -1, -2, 0, -1, -2], [0, 0, -2, 0, -2, 0, -2, 0, 1, -2, 2, 0, -4, -2, -2, -2, 0, -1], [0, -2, -2, 0, -2, 2, -2, 0, 2, 0, -6, -2, 0, -2, 0, 2, -1, -2], [0, -1, -2, -1, -2, 0, -3, 0, -1, -1, 5, -3, 0, -1, 5, 4, 0, 5], [2, 0, 0, 2, 0, -2, -2, -2, -1, 2, 2, -2, 0, 0, -2, 0, 2, -1], [5, -3, 1, 2, -3, -3, -1, 2, -2, -2, -2, 1, 0, 1, -2, -3, 1, -2], [1, 3, 4, 4, -2, 1, 0, 4, -3, 3, -4, 3, 8, -2, 1, 2, -3, -1], [0, 3, -3, 0, -2, -3, -1, -1, -2, -2, 1, 3, 1, -3, -2, 4, -1, 2], [5, 2, 0, -1, 1, -2, 2, -1, 4, -4, 3, -2, -4, 2, 2, -2, 3, -2], [0, 3, -1, -8, 1, -3, -1, 0, 3, 0, 3, -3, -3, 1, 0, 2, 3, 3], [0, 1, 0, 0, 2, -3, 2, 0, -3, 1, -1, 3, 2, 0, 1, 2, 1, 3], [5, -2, -1, 1, 1, -2, -1, -2, 1, -1, 7, 0, -4, -1, -1, 1, 3, 1], [0, 2, 0, -1, 4, -3, 3, 2, 0, 0, -2, 2, 0, 3, -2, -4, 1, -2], [-2, 0, -2, -3, 2, 4, 2, 4, 0, 4, -4, 4, 4, 2, -4, -6, -4, 0], [-7, 2, -2, 0, 4, -2, -2, 0, 2, 2, 1, 2, 0, 4, 0, -4, -4, 0], [-3, -1, 0, 1, -1, -1, -4, -1, 1, 1, 0, 1, 0, 2, -1, 6, -2, -1], [4, -3, 2, 0, -2, 1, 0, -2, 1, 1, -1, 1, -4, -2, -1, 6, -1, 1], [-2, -2, 1, -4, -1, 0, -1, -2, 0, 3, 0, 0, 3, -1, -3, -2, 0, 0], [3, -3, -1, -1, -1, -3, -3, 2, 0, 0, -4, 1, 0, 3, 0, 7, 0, 0], [3, 7, -2, 2, 0, -7, -6, -2, -5, -5, 6, 1, 0, 2, 1, 2, 1, 1], [-2, -1, -3, -4, 4, 4, -2, 1, 2, -2, 3, -1, -1, 6, -2, -6, 1, 2], [1, 2, 0, -6, 2, 2, 0, 2, 2, -2, 0, -1, 4, 4, 1, -4, -4, -1], [0, 2, -1, 0, -2, 0, -1, -1, -1, 1, 4, 0, -3, -1, 1, -2, 4, 1], [6, -7, 2, 10, -6, 1, 4, -2, -2, -1, -1, 1, -4, -2, 1, 6, -1, -2], [0, -6, 5, 0, 5, 4, 3, 2, 6, 1, -6, -4, -1, 5, 1, -6, -2, -6], [10, 0, -2, -1, 0, -4, -2, -2, 0, -4, 4, -4, -4, 2, 4, 2, 4, 0], [0, 0, -2, 0, -2, -4, -6, 4, -1, -2, -6, 4, 0, -2, -2, 2, 0, 1], [-8, 5, 3, -1, 0, 3, 5, 5, 2, 0, -3, 3, 7, -3, 0, -4, -3, 2], [-6, 1, -3, 4, 4, -3, -7, -3, -2, -2, 3, -3, -3, 3, 2, 0, 3, -2], [3, -2, 4, -4, 0, 2, 4, 0, 2, 2, 1, -8, 0, 0, 4, 0, 6, 4], [0, -3, 5, 0, 5, -1, 1, 4, -1, 6, -5, 1, 9, 5, 6, 4, -9, 1], [-1, -2, -3, 3, -3, 6, 1, 0, -1, -1, 1, 2, 0, -7, 1, 7, -5, 1], [3, -4, 2, 0, -2, 0, 2, 2, 0, 0, -5, 2, 0, 0, -4, -2, 2, -4], [0, -3, 1, 0, 2, 5, 5, -3, 4, 0, 11, -5, -5, 1, 0, -6, 6, -4], [-3, -4, 4, 6, 4, 4, 0, 2, -2, 2, -9, 2, 4, 2, 2, -2, 0, -2], [-3, -6, 4, 0, 0, 2, 0, 0, -6, 6, -7, 4, 4, -4, 0, 4, -10, 0], [0, -2, 0, 0, -2, 2, -4, 2, -2, 0, -8, -2, 4, 0, 0, 4, -4, 2], [-6, -1, 5, 7, -2, 4, 6, 1, -4, 4, -11, 5, 5, 0, 0, 0, -2, 0], [0, 7, 2, 0, 0, -1, 4, -4, -2, 1, 1, 1, 4, 2, 1, 2, -1, 2], [-3, -3, -1, 0, -1, -1, -1, -2, -2, 2, -2, -1, 8, -3, 0, 3, -1, 0], [8, 0, 0, -1, -2, 3, 3, 0, -2, -2, -4, 0, 0, -5, 0, -4, 3, 0], [5, -5, 4, 6, -2, 5, 8, 2, 1, 1, -5, -2, 0, -4, -2, -4, -3, -2], [0, 6, -1, 0, -1, -6, -1, 0, 2, -1, 2, 6, 1, -1, -1, 2, 4, -2], [2, 3, -2, -6, 2, 3, 4, 6, 1, -3, -3, 3, 2, 2, 3, -4, -3, 1], [-10, -4, -9, -4, 5, -2, -7, -2, 0, -1, 2, -2, -5, 9, 1, -4, 2, 0], [-1, -3, -5, 2, 1, -1, -1, 4, 0, 0, -4, 5, 0, 1, 0, 3, -7, 0], [3, 7, 0, -4, 2, -7, -8, 4, 5, -5, 5, 2, -2, -2, -2, 2, 1, 2], [0, -10, -1, 0, 4, 4, 3, 1, 5, -3, 4, -4, -9, -1, -3, -8, -4, -5], [-9, 1, -6, -5, 3, 3, -8, 3, 1, -1, 0, -3, 0, 4, -3, -4, 2, 3], [0, -4, 6, 0, 2, 4, -2, 4, 2, 4, -2, -4, 2, 6, 4, -4, -6, -2], [-2, 6, -4, -7, 6, 2, 0, 6, 2, 0, -2, 2, 0, 4, 0, -4, -2, 2], [0, 9, 5, 0, 3, -1, 9, -6, 3, -4, 5, 1, -3, 5, -4, -8, 13, -3], [-5, 0, -4, 0, 0, 0, -4, 0, 0, 0, 5, 2, -4, 0, -6, 0, 4, 6], [7, -1, 4, 2, -2, 5, 4, -2, 1, 1, -6, -3, 0, -4, -1, -2, 1, -1], [4, -1, 0, 6, 0, 0, 3, 2, -1, -1, 3, -5, 0, -1, 5, 10, -5, 5], [-14, -6, -8, -5, 4, -2, -8, -4, 0, -4, 2, -2, -6, 8, 4, -2, 2, 0], [-12, 0, -6, -6, 2, -4, -2, 0, 6, -2, 4, -4, -6, 6, 2, 0, 4, 6], [0, -2, 6, 8, -6, 5, 9, 0, -4, -4, 8, 0, 0, -9, -6, -2, 2, -6], [-5, 2, 3, -3, 1, -6, -5, -2, 1, 1, 7, -4, 0, 7, 1, -3, 5, 1], [3, 1, -1, -4, 5, -5, -5, 0, 4, 4, -8, -3, 0, 9, 4, 3, -1, 4], [-6, 4, -1, -3, -8, -1, 4, -1, -3, 3, -2, 2, 9, -4, -3, 4, -5, 3], [5, 2, -4, 4, -2, -2, -4, 0, 0, 0, 9, -4, -4, -2, 0, 2, 8, 0], [0, 6, 0, 0, -4, -6, -8, 4, -6, -2, 2, 6, 4, 0, -2, 4, 2, 6], [-10, -12, 0, 3, -6, 4, 4, -2, -4, 2, -4, 4, 0, 0, -2, 0, -4, -4], [-9, 5, 0, -6, 10, -1, 8, 2, 3, 3, 2, 1, 0, 4, -1, -6, 3, -1], [18, -2, 4, 1, -8, -2, 8, 0, 0, 0, 2, -2, -2, -4, 0, 6, 2, 0], [0, 3, -4, 0, -4, 1, 0, -4, -4, 1, -1, -1, 4, -4, 1, 8, 3, 4], [0, -2, 2, 0, 0, 6, 2, -2, 6, -4, 6, -6, -8, 2, -4, -10, 7, -6], [2, 0, 5, 4, -1, 2, -1, -2, 2, -3, -2, 2, 5, -5, 3, 0, -2, 2], [9, 9, 1, 1, 3, -1, 1, -4, -2, -2, -6, 1, 0, -1, -2, -11, 2, -2], [1, 3, 0, 2, 2, -3, 8, 4, -5, -5, 6, 3, 0, -2, -3, 0, 3, -3], [0, 9, -3, 0, -2, -3, 5, -7, 2, -4, -9, 3, -7, -3, -4, -4, 6, -2], [-1, -6, -4, 0, 4, -6, -4, -6, 4, -4, 3, -4, -6, -2, 0, 2, 8, 0], [0, 3, -6, 0, -4, -5, -4, 0, -1, -3, 13, 5, -6, -6, -3, 0, 7, 1], [0, 4, -1, 0, -1, 2, 5, -6, 4, 1, 4, -2, -7, -1, 1, -6, 0, -4], [4, -4, -5, -8, 4, -3, -14, 1, -1, 1, 10, -4, 5, -4, -1, 4, -4, 1], [0, -3, -3, 0, -1, 9, 7, -8, 3, 0, -11, -9, -5, -3, 0, -2, 3, -3], [-5, 5, 2, -3, 1, 7, 0, 7, -1, 1, -12, -3, 10, 6, 5, -6, -6, -5], [5, -12, -5, -2, 3, 2, 1, -8, -1, 1, 7, -8, -6, 1, 3, -1, 2, -3], [-18, -5, 3, 12, -6, 11, 15, 9, 0, 6, -11, 11, 9, -3, -6, -6, -11, 0], [14, 0, -1, 8, -7, -6, 9, 2, -2, -5, 6, -6, -3, 1, 5, 2, 6, -2], [0, 1, 9, 0, 6, -3, 1, 5, 6, 0, -17, 3, 5, 9, 0, -4, 2, -6], [-6, -6, -4, 6, -4, 3, 1, 4, -2, -2, 2, 8, 0, -5, -4, 4, -8, -4], [0, -5, 5, 6, -9, -1, 7, -2, -5, 0, 1, -1, -3, -5, 0, 8, 1, -5], [-5, 7, 1, -8, 9, 5, 9, -2, 4, 4, -4, 1, 0, -1, -2, -3, 5, -2], [-5, -12, -8, -2, 6, 4, -12, -2, -4, 4, 2, -9, -4, 0, 1, 0, 2, -1], [-7, 7, 2, 1, -7, 5, 0, 7, -7, 7, -4, 7, 2, -2, -5, 2, -8, 5], [4, -5, 3, 4, -3, 1, -1, -4, -3, -4, -1, 1, 5, -3, 4, -2, -1, -3], [4, 9, 7, 3, 0, 3, 5, 5, -4, -4, -3, 3, 9, -7, 4, -2, -3, -4], [-3, -6, 5, -1, 5, 6, -3, 4, -5, 5, -7, 6, 12, 1, -1, -1, -11, 1], [5, -1, -6, 2, -2, 5, 2, 2, -3, -3, -5, 6, 0, -8, 0, -6, 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