
# q-expansion of newform 400.2.u.f, downloaded from the LMFDB on 24 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 = 400
weight = 2
poly_data = [16, 168, 688, 84, 1473, 786, 342, -6, 93, -24, 13, -3, 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], [[112273499910, 521633846125, 442357548822, 677052260712, 224325622533, 129671476953, 3519097812, 25045472937, -4900209927, 3020952123, -576792708, 260325772], 438461548785], [[178985069048, 2016280256236, -2849443497756, 3903365483097, 2207600053386, 1125752172738, -127593738606, 339987222837, -97783045044, 48283397793, -11783661403, 3457672033], 1753846195140], [[391154012184, 2657911506232, 489856121406, 8668716529509, 4299712819554, 1832724711006, 23885891676, 481102476429, -119576878026, 67292419131, -15433734165, 5369632561], 1753846195140], [[-273874062464, 21091602933640, -5927387741190, 28540425731691, 15333778688490, 7906799179014, -814213247460, 2276267517771, -633522002730, 314418060009, -74034744959, 22373133631], 3507692390280], [[-5593310307856, -23515293707092, -6601997100420, -46013446580871, -23654460062430, -10403352643614, 95827068690, -2749867795311, 696924743820, -382317412959, 87105581309, -29615632063], 3507692390280], [[-17878470653712, 67498588603448, -80865862575282, 96648914614131, 56181193046322, 34599514729674, -6581106786312, 10803833121471, -3218484118998, 1484218111509, -361617020679, 99321102659], 8769230975700], [[2898411243400, 32659532805012, -13230315931086, 63421806854271, 34765432275066, 16674062237514, -1255570461996, 4786319147691, -1308046874814, 666492738129, -157422019691, 48894251523], 3507692390280], [[-37003359759088, -90978016936148, -39698554795488, -260834151846351, -140798466578802, -55756593480954, -1748332186158, -14810975764191, 3656650156068, -2090376253539, 478101391109, -167923324379], 8769230975700], [[77595685775224, 336336630781804, -94750243258356, 719350254826743, 399600830367126, 182680683235422, -10432981854546, 52158480741663, -14065746308484, 7294073305527, -1717504490237, 542891618887], 17538461951400], [[26266096925344, 363489467079144, -43808899935726, 796406422106523, 426485463435546, 197679060702342, -9699921955716, 54915725330643, -14590245523914, 7631251682697, -1777970044487, 572643183207], 17538461951400]]

hecke_ring_character_values = [[351, [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], [101, [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], [177, [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0]]]
aps_data = [[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, -1, 1, 1, 0, -2, 0, 1, 1, -1, -1, -1], [1, -1, 0, 0, 0, 1, 1, 0, 0, 0, -1, 1], [0, 1, 1, 0, -1, -1, 0, -1, 0, 0, 0, -1], [1, 0, 0, -1, 0, 1, -1, 0, -1, 1, -1, 0], [2, 0, -1, -1, 0, 2, 3, 0, -3, 0, 0, 0], [2, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, -1], [1, 0, 1, 1, 0, -1, 1, 1, 0, -2, -1, -1], [-2, -1, -1, -1, -2, 2, -2, -1, -3, 1, 1, 2], [0, 0, -2, -2, 1, 0, 2, 0, -2, 0, 0, 0], [-1, -1, 1, 2, 0, -1, -4, 0, 1, -1, 1, 0], [0, -1, 1, 2, 1, 0, 4, 2, -1, -2, 0, -1], [-6, -1, 1, 2, -2, -2, -2, -1, 0, -1, 0, 0], [-1, 0, -1, -2, -1, 2, -1, -1, 0, 1, 1, 2], [2, -2, 0, 0, -2, 0, 2, -1, 2, 0, -1, 0], [-2, 3, -2, -3, -2, 6, -2, -4, -4, 2, 2, 2], [-3, 1, -1, 0, 1, 1, 0, -2, 3, 0, 0, -2], [0, -1, 0, 1, 2, -2, -6, 1, 8, -2, -2, 0], [-3, 0, 1, 2, 1, -2, -3, 3, 4, -1, 1, -2], [1, 1, -1, 2, 3, -5, 2, 2, 1, -4, -2, -2], [1, 0, -2, 1, -2, 4, 1, -1, -4, 2, 3, 4], [4, 0, 2, 0, -1, 2, -4, 0, 4, 2, 0, -2], [2, 0, -2, -1, 1, 9, -1, -1, -3, 2, 1, 1], [-2, 3, -2, -4, 1, 4, -2, -1, -3, 2, 3, 4], [4, 0, 0, 1, -1, 1, 1, -2, 0, -2, 0, 0], [-3, -2, 0, 1, -2, 0, -3, -3, -2, 0, -3, 0], [3, 0, 1, 0, 0, 7, 7, 1, 0, 1, 1, -1], [0, 3, -1, -2, 1, -2, 4, 2, 1, -2, 0, -1], [-2, 4, -4, -3, -1, 8, -1, -2, -7, 0, 2, 1], [0, -1, 3, 0, -3, -11, 0, -1, 0, 0, 0, -1], [6, -1, -3, -1, -2, 5, 8, 1, -6, -3, -2, 1], [2, -1, 1, -1, 0, -7, -4, 0, 7, 2, -2, 0], [-5, 6, -3, -3, 0, 1, -3, -3, 2, 6, 3, 3], [3, 2, 0, 3, -3, -5, -5, 1, 0, 1, 2, -2], [8, -5, 5, 2, -2, 6, 6, 1, 0, 1, 0, 0], [6, -6, 8, 5, -5, -1, -1, 2, 0, 2, 2, -2], [-5, 4, -1, -2, -3, 5, 1, -6, -2, 1, 5, 3], [-10, 2, -1, -2, -1, -5, -8, -2, 4, 3, 4, 1], [-1, 3, 2, -2, -4, -1, -2, -3, -1, 4, 2, 1], [6, -6, 2, 3, -4, -4, 6, 0, 2, -2, -4, -4], [2, -2, 1, -1, 4, -6, 1, 2, 3, 0, -4, -4], [1, -2, 1, -1, -2, 4, -1, -4, -2, 5, -1, 2], [2, -2, 1, 1, -1, 8, 8, 2, 0, 2, -1, 1], [-4, 0, 1, -3, 1, -2, -4, -1, -4, -1, -3, -2], [5, 1, -2, -2, 2, -5, -5, 2, 0, 2, -1, 1], [5, 2, -3, 3, 6, -1, 3, 3, -2, -6, -3, -3], [-2, -3, 5, 0, -5, -9, 0, -3, 2, 0, 0, -3], [0, -5, 1, 2, 1, -5, 2, 5, 2, -4, -2, 1], [8, 0, -1, -5, -1, 2, 8, -5, -4, 1, -3, 2], [1, -1, 5, 5, -1, -2, -2, 1, 4, -1, -2, -1], [3, 0, 1, 3, 2, 1, 3, 3, 0, -6, -3, -3], [1, -1, -1, 0, 0, -3, -4, 0, 3, 1, -1, 0], [-11, -2, 3, 4, -4, -1, -1, -3, 0, -3, 1, -1], [6, 3, -2, 3, -3, 8, 8, -3, 0, -3, 1, -1], [2, 3, -2, -2, 3, -11, -2, 6, 8, -4, -2, -3], [-9, 3, 1, 1, 1, 0, -14, -3, 8, 3, 6, 3], [-4, 0, -1, 2, -1, 2, -4, -2, -8, 1, 0, 2], [-1, -3, 2, -2, -4, -3, -2, -5, -1, 4, 2, -1], [-15, 4, -2, 0, 0, -1, -8, -4, 0, 2, 8, 6], [-14, 3, -9, -5, -2, -1, -8, -3, 2, -1, 6, 7], [-7, -2, 3, 1, -1, -8, -8, 3, 0, 3, 1, -1], [6, 2, -2, -2, 2, 2, 2, 4, 0, 4, 0, 0], [-3, 7, -3, -3, 0, 12, -3, -2, 0, 6, 3, 4], [1, 4, 1, -3, 2, -10, -3, 4, 8, -3, -1, -2], [12, 0, -2, -4, 1, 10, -2, 0, 2, 2, 0, -2], [-4, -1, -2, -1, 4, -6, 4, 1, -2, -2, -2, 0], [0, -5, 0, 2, -3, 17, 2, -6, -14, 6, 0, 3], [11, -4, 4, 7, 0, -8, 11, 5, -4, -4, -3, -8], [3, 4, -9, -3, 3, 6, 6, -1, 0, -1, -5, 5], [-11, -4, -3, 2, -7, 6, -11, 1, 2, 3, 7, 6], [-2, 3, -4, -2, -1, -13, 2, -2, 14, 0, 2, 1], [6, 3, 1, -1, -10, 13, 12, -3, -18, 5, 6, 1], [6, -4, 3, 0, -3, 0, 0, 2, -6, 0, 0, 2], [4, -2, 2, 3, 0, -4, 4, -2, -2, -2, -6, -4], [-8, 0, 2, 2, 1, -8, -22, 0, 22, 0, 0, 0], [-11, -1, -5, 4, 9, -5, 4, 4, 15, -8, -4, -4], [-1, -2, -2, 3, -4, 4, -1, -3, 0, 2, 1, 4], [1, 4, -6, -3, 3, 5, 5, 1, 0, 1, -2, 2], [5, -5, 7, 4, 4, -11, 0, 8, 7, -3, -5, -4], [0, 0, 6, 6, -3, 0, 2, 0, -2, 0, 0, 0], [-13, 3, -2, -8, 1, 4, -13, -6, 9, 2, -2, 4], [-6, 2, -2, -2, 7, 0, -20, -2, 16, 2, 4, 2], [5, 2, 0, 2, -6, 13, -8, -2, 4, 0, 4, 4], [1, -3, 3, 1, -2, 6, 1, 6, 0, -2, -1, 4], [-1, 4, 1, -2, 2, 13, 13, 5, 0, 5, 5, -5], [9, -1, 4, -3, 3, -6, -6, 6, 0, 6, 3, -3], [2, 8, -8, -8, 8, 4, 4, 4, 0, 4, 0, 0], [11, -11, 2, -2, -4, 13, -2, -5, -13, 4, 2, -1], [5, -2, 5, -4, -1, 7, 3, -2, -6, 7, -5, 1], [0, -1, 5, -1, -4, -9, 4, 1, -2, 5, -2, -7], [-19, 2, -4, -3, -2, 8, -19, -3, -2, 4, 5, 8], [-5, -2, -1, 2, -5, 13, -1, -10, -8, 5, 5, 5], [5, -4, 7, 4, 5, 1, 13, 10, -6, -5, -5, -5], [11, 3, -8, -6, 6, -1, -1, -2, 0, -2, -5, 5], [8, 4, -3, -2, 1, 6, 8, -4, -20, 3, 2, 6], [0, -2, 0, 0, -2, 11, 13, -4, -9, 4, 0, 2], [-10, -2, -2, 4, -4, 4, -10, -3, -2, 2, 1, 4], [14, -8, 7, 2, -5, 2, 2, 0, -12, -4, -2, -4], [-1, 1, -1, 1, 1, 2, -18, 2, -3, -3, 1, -1], [-4, -2, 0, -3, -2, 0, -4, 2, -14, 0, 2, 0], [3, -4, 3, -3, 3, 10, 10, -1, 0, -1, -1, 1], [-2, -3, 0, 4, -1, 5, -20, -2, -4, 0, 2, 1], [-6, -1, -1, 2, -7, -6, -5, 1, 7, -4, -2, 2], [-12, -2, 4, 5, 2, -8, -12, 6, 10, -4, -2, -8], [0, 5, 3, -4, -7, -5, -4, -5, -4, 8, 4, 3], [-1, 0, 13, 0, -1, -3, -23, -2, 4, 1, 1, 1], [4, 1, -7, -11, 11, -13, -13, 1, 0, 1, -6, 6], [9, 8, -5, -8, 8, 1, 1, -1, 0, -1, 3, -3], [-3, 7, -7, -9, 9, 9, 9, 2, 0, 2, 0, 0], [-1, -7, 10, 4, -4, -19, -19, 4, 0, 4, 3, -3], [-6, 4, -2, -4, -6, 3, -5, -12, 3, 6, 6, 6], [21, -2, 9, 4, 4, 10, 2, 2, 2, 3, -4, -7], [-2, -4, 4, 2, -1, -16, -2, 4, 10, -2, -8, -6], [8, -4, 6, 0, -1, -10, -8, 4, 16, 2, -8, -10], [0, -5, 8, 6, 1, -7, 14, 2, 6, -2, 0, -1], [16, 8, -8, -4, 4, 12, 12, -4, 0, -4, 0, 0], [8, 2, 3, 0, 5, -6, 8, 0, 14, -3, -6, -6], [-12, 9, 2, -6, 11, -4, -12, 3, 19, -2, -1, -4], [3, -12, 3, 7, -2, 12, -3, -4, -10, 7, -3, 2], [6, -10, 3, 0, -3, 10, 0, 4, -6, 0, 0, 4], [-2, -2, 6, -5, -11, 3, -5, -11, -3, 10, 5, -1], [-1, 1, 1, 5, 4, -18, 5, 0, 6, -10, -5, -10], [-22, 2, 2, 6, -5, -12, -16, -2, 12, -2, 4, 6], [-8, 13, -16, -13, -8, 16, -12, -16, -8, 8, 8, 8], [14, -2, -1, 3, -3, 0, 0, -2, 0, -2, -3, 3], [2, -1, -5, -1, 0, 11, -4, 0, -11, 2, -2, 0], [-2, -4, 2, 2, 7, -14, 10, 4, -2, -4, -8, -4], [-6, -6, -1, -6, -5, 0, -6, -2, 0, 12, 6, 10], [-8, -2, 5, 4, -3, -15, -4, 2, 8, -1, -4, -3], [-14, 10, -1, 2, 3, 0, 2, 2, 16, -4, -2, -2], [-20, 0, -2, -10, 10, 8, 8, 0, 0, 0, -2, 2], [-15, 8, -3, 6, -6, -3, -3, -1, 0, -1, 5, -5], [1, -2, 2, 7, -7, -3, -3, 3, 0, 3, 0, 0], [16, 7, 5, -1, -6, 12, -1, -3, -17, 2, 1, -1], [-12, 3, 6, -1, -6, -10, -14, -3, 8, 10, 6, -4], [8, 7, -7, 1, 8, 10, 1, 1, -7, -2, -1, -1], [-1, -2, 5, 7, 4, -10, -9, 8, 6, -9, 1, -4], [-8, 2, 0, 4, 2, 0, -8, 9, -18, 0, 9, 0], [-8, 0, -4, 6, -2, 1, 3, -4, 1, -4, 8, 2], [5, 2, -1, -5, -4, 15, -5, -11, -10, 10, 5, -1], [3, -4, 5, 6, 1, -10, 3, 11, -18, -5, 1, -10], [-16, 1, -5, 0, 5, -13, 0, 5, 16, 0, 0, 5], [2, -14, 1, 2, 1, 0, 2, 4, 0, -4, -2, 0], [-1, 2, -5, 1, -3, 10, -1, -2, 2, 5, 8, 10], [2, 4, -13, -6, 7, 14, -6, 0, -8, 12, 6, 12], [6, 0, -5, -9, -5, 10, 6, -11, -20, 5, -1, 10], [-8, 7, -7, -6, 6, -6, -6, 7, 0, 7, 0, 0], [-3, 2, 3, 0, -1, -7, -13, -2, 8, -1, 3, 1], [-6, -5, 10, 7, -4, -16, -6, -8, 20, 2, 6, 4], [6, -2, -11, 5, 16, -5, 5, 12, -1, -10, -5, 2], [-16, -7, 2, 1, 12, -38, -10, 7, 24, -6, -14, -8], [-16, -5, 12, 15, 0, -28, -4, 5, 14, -8, -10, -2], [7, -6, -5, 1, 2, -2, 7, 4, 0, 3, -7, -2], [1, 8, -7, -6, 6, 13, 13, -1, 0, -1, 1, -1], [-11, -2, -1, 3, 4, -25, 3, 5, 14, -6, -3, -1], [-2, 2, 3, 6, -5, 7, -8, -2, 4, -1, 4, 5], [-4, -5, 4, -2, -1, -8, -4, -3, -3, -4, -11, -8], [8, -6, 2, 3, 5, -18, -11, 10, 13, -2, -8, -5], [-7, 13, -9, -12, -6, 9, 14, -12, -3, 5, 7, 6], [26, 4, -3, -6, 1, 6, 26, -2, -20, 3, 4, 6], [6, -1, 12, 6, 11, -24, 6, 15, 29, -12, -9, -24], [18, 7, -8, -3, -4, 44, 14, -7, -28, 2, 14, 12], [7, 0, -4, -3, -4, 8, 7, 3, -8, 4, 11, 8], [-19, -1, -5, -2, 3, -7, -2, 0, 17, 4, 2, 4], [36, -6, 1, -2, 5, 15, 8, 6, 4, -3, -12, -9], [5, 4, -13, -6, 3, -1, 25, 6, -2, -1, -5, -3], [-2, 4, -8, -9, -4, 16, -2, -16, 6, 8, 0, 16]]
