
# q-expansion of newform 405.4.e.m, 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
    R = PolynomialRing(QQ, "x")
    f = R(poly_data)
    K = NumberField(f, "a")
    betas = [K.gens()[0]**i for i in range(len(poly_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 = [1, -1, 1]

# The basis for the coefficient ring is just the power basis
# in the root of the defining polynomial above.
hecke_ring_character_values = [[326, [0, -1]], [82, [1, 0]]]
aps_data = [[5, -5], [0, 0], [0, -5], [-9, 9], [8, -8], [0, -43], [-122, 0], [-59, 0], [0, 213], [-224, 224], [0, 36], [206, 0], [0, -413], [392, -392], [311, -311], [-377, 0], [0, -337], [-40, 40], [0, -348], [62, 0], [-1214, 0], [294, -294], [-534, 534], [-810, 0], [928, -928], [996, -996], [0, 433], [-1686, 0], [656, 0], [0, -1018], [-1361, 0], [0, -1911], [-654, 654], [0, 733], [0, 1126], [2546, -2546], [0, -1223], [3176, 0], [0, -132], [993, -993], [-3101, 0], [2846, 0], [-3080, 3080], [0, -2588], [-1335, 0], [-5204, 0], [0, -1637], [4480, -4480], [3736, -3736], [0, 1380], [-2904, 0], [0, 5966], [3218, -3218], [6123, 0], [0, 1398], [3211, -3211], [4018, 0], [2314, 0], [4347, -4347], [-1551, 1551], [0, -4380], [0, -5049], [5428, 0], [0, 18], [2116, -2116], [-4415, 4415], [-3480, 3480], [0, -6322], [0, 10034], [2510, -2510], [-3726, 3726], [-10710, 0], [2160, -2160], [0, -3394], [9031, 0], [0, 10305], [3480, -3480], [3706, 0], [0, -2679], [0, 12499], [0, 928], [-7570, 7570], [-2460, 0], [1648, 0], [-15826, 15826], [-12774, 12774], [8875, 0], [-11524, 11524], [8544, -8544], [0, -2523], [-2902, 0], [-4362, 4362], [5723, 0], [0, 13339], [0, 19637], [-5416, 0], [0, 6110], [20375, 0], [-19010, 0], [3288, 0], [-3256, 3256], [213, 0], [0, -17388], [4353, -4353], [0, 1924], [16832, 0], [2106, -2106], [4694, 0], [0, -13226], [13291, -13291], [0, -6040], [-23, 0], [0, 3018], [9439, -9439], [-9800, 0], [-23442, 23442], [0, -31308], [712, 0], [0, -31478], [16121, -16121], [0, 19160], [13422, -13422], [-25905, 25905], [-9246, 0], [-25039, 25039], [32930, 0], [-1882, 1882], [-1962, 0], [13741, -13741], [0, 32458], [19612, 0], [0, -36736], [0, 3746], [5725, 0], [0, -37323], [0, 24586], [-3078, 0], [0, -41038], [0, -9362], [-45115, 0], [-13512, 0], [4530, -4530], [0, -30884], [12088, 0], [-14112, 0], [-3860, 3860], [-2862, 2862], [32534, -32534], [0, 500], [-36595, 0], [0, -40143], [16414, 0], [-33478, 0], [0, 3633], [-20466, 20466], [15074, -15074], [-36848, 0], [-35174, 35174], [-10092, 0], [0, 12910], [48060, -48060], [6316, 0], [0, -7760], [22437, 0], [0, 3506], [30648, -30648], [19232, 0], [-38415, 38415]]
