
# q-expansion of newform 1728.3.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 = 1728
weight = 3
poly_data = [1, 0, -1, 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], 1], [[0, 0, 0, 1], 1], [[-3, 0, 6, 0], 1], [[0, 6, 0, -3], 1]]

hecke_ring_character_values = [[703, [-1, 0, 0, 0]], [325, [-1, 0, 0, 0]], [1217, [1, 0, 0, 0]]]
aps_data = [[0, 0, 0, 0], [0, 0, 0, 0], [0, 6, 0, 0], [0, 5, 0, 0], [6, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, -6], [0, 0, 0, -5], [0, 0, 6, 0], [0, -36, 0, 0], [0, -26, 0, 0], [0, 0, -5, 0], [0, 0, 0, 0], [0, 0, 0, -4], [0, 0, 6, 0], [0, 60, 0, 0], [18, 0, 0, 0], [0, 0, -15, 0], [0, 0, 0, -15], [0, 0, 0, 0], [25, 0, 0, 0], [0, 31, 0, 0], [120, 0, 0, 0], [0, 0, 0, 30], [-85, 0, 0, 0], [0, -156, 0, 0], [0, -125, 0, 0], [-150, 0, 0, 0], [0, 0, -20, 0], [0, 0, 0, -6], [0, 110, 0, 0], [-156, 0, 0, 0], [0, 0, 0, 30], [0, 0, 0, -35], [0, -24, 0, 0], [0, -31, 0, 0], [0, 0, 4, 0], [0, 0, 0, -59], [0, 0, 30, 0], [0, 120, 0, 0], [192, 0, 0, 0], [0, 0, -65, 0], [0, 0, 30, 0], [-325, 0, 0, 0], [0, 30, 0, 0], [0, 259, 0, 0], [0, 0, 0, -5], [0, -310, 0, 0], [-360, 0, 0, 0], [0, 0, -40, 0], [0, 0, 0, -60], [0, 0, 60, 0], [-131, 0, 0, 0], [-216, 0, 0, 0], [0, 0, 0, -36], [0, 0, 60, 0], [0, -414, 0, 0], [0, 317, 0, 0], [0, 0, -100, 0], [0, 0, 0, -60], [0, 0, 0, -20], [0, 270, 0, 0], [0, 0, 0, 84], [0, 0, -90, 0], [85, 0, 0, 0], [0, 120, 0, 0], [0, 0, 0, 5], [-325, 0, 0, 0], [300, 0, 0, 0], [0, 0, 115, 0], [0, 0, 0, 84], [0, 0, 90, 0], [0, -185, 0, 0], [0, 0, 11, 0], [0, 0, 0, 25], [0, 0, -36, 0], [0, 354, 0, 0], [0, 0, 124, 0], [0, 0, 0, 60], [-539, 0, 0, 0], [-306, 0, 0, 0], [0, 0, 5, 0], [0, 0, -30, 0], [250, 0, 0, 0], [0, 22, 0, 0], [-540, 0, 0, 0], [0, 0, 0, 30], [-230, 0, 0, 0], [0, 78, 0, 0], [0, -55, 0, 0], [-150, 0, 0, 0], [0, 0, -60, 0], [0, 785, 0, 0], [618, 0, 0, 0], [0, 0, 0, 120], [0, 0, -126, 0], [0, 366, 0, 0], [0, 0, 0, -150], [0, 0, 0, 21], [0, 0, 125, 0], [0, 0, 0, 81], [0, 690, 0, 0], [-840, 0, 0, 0], [0, 0, 0, -210], [0, 0, 0, 155], [385, 0, 0, 0], [930, 0, 0, 0], [0, 0, 0, -96], [0, 0, 60, 0], [-346, 0, 0, 0], [0, -595, 0, 0], [0, 0, 21, 0], [0, 0, 0, -114], [0, 0, 0, 135], [0, 713, 0, 0], [0, 0, 0, 0], [0, 0, 0, -84], [0, 0, 120, 0], [0, -960, 0, 0], [-132, 0, 0, 0], [0, 0, -195, 0], [445, 0, 0, 0], [0, -990, 0, 0], [-1260, 0, 0, 0], [0, 0, 0, 20], [0, -246, 0, 0], [0, 0, 25, 0], [0, 0, 240, 0], [0, 770, 0, 0], [0, 0, 220, 0], [0, 0, 0, -140], [0, 0, 114, 0], [0, 1099, 0, 0], [0, 0, 81, 0], [0, 0, 0, -30], [-1009, 0, 0, 0], [0, 360, 0, 0], [0, 0, 0, 99], [0, 660, 0, 0], [0, 0, 0, 240], [0, 0, 0, -100], [0, 1242, 0, 0], [0, -955, 0, 0], [-90, 0, 0, 0], [0, 0, 35, 0], [0, 0, -300, 0], [0, 0, -9, 0], [0, 0, 0, 120], [0, 0, 0, -25], [0, 0, -66, 0], [0, 0, 199, 0], [0, 0, 0, 90], [0, 0, 0, -55], [0, 0, -96, 0], [0, 0, 0, -211], [0, 0, 120, 0], [0, -1006, 0, 0], [0, 0, 0, -60], [565, 0, 0, 0], [0, 294, 0, 0], [30, 0, 0, 0], [0, 0, 0, -174], [0, -305, 0, 0], [1194, 0, 0, 0], [0, 0, 0, -24], [0, 0, 246, 0], [0, 511, 0, 0], [0, 0, -204, 0], [119, 0, 0, 0], [0, -1530, 0, 0], [96, 0, 0, 0], [0, 0, -300, 0], [0, 0, 330, 0], [95, 0, 0, 0], [0, 1711, 0, 0], [0, 0, 0, 120], [0, 0, 0, -140], [0, 222, 0, 0], [0, 475, 0, 0], [0, 0, -80, 0], [0, -1625, 0, 0], [-1032, 0, 0, 0], [0, 0, -276, 0], [0, 0, 0, -174], [0, 0, 60, 0], [0, 1668, 0, 0], [0, 0, 39, 0], [0, 0, 0, -301], [683, 0, 0, 0], [0, 0, 270, 0], [-1955, 0, 0, 0], [-1380, 0, 0, 0], [0, 0, 0, -25], [0, 1074, 0, 0], [-450, 0, 0, 0], [0, 0, 0, -210], [577, 0, 0, 0], [0, 0, -109, 0], [0, 0, 0, -90], [0, 0, 180, 0], [0, -2226, 0, 0], [0, -787, 0, 0], [0, 0, 155, 0], [-373, 0, 0, 0], [804, 0, 0, 0], [0, -900, 0, 0], [0, -871, 0, 0], [750, 0, 0, 0], [0, 0, 0, -330], [0, 0, 0, -455], [-1465, 0, 0, 0], [0, -1674, 0, 0], [0, 1615, 0, 0], [-2010, 0, 0, 0], [0, 0, -420, 0], [851, 0, 0, 0], [0, 1990, 0, 0], [0, 0, 0, 270], [0, 0, 216, 0], [0, 360, 0, 0], [0, 0, 95, 0], [0, 1414, 0, 0], [0, 0, 0, 180], [0, -1150, 0, 0], [-150, 0, 0, 0], [0, 0, 145, 0], [0, 0, 0, 186], [0, 0, -210, 0], [0, 535, 0, 0], [414, 0, 0, 0], [0, 0, 65, 0], [0, 0, 0, -40], [0, 1214, 0, 0], [0, 0, 0, 30], [0, 0, 0, 35], [0, 0, 30, 0], [359, 0, 0, 0], [0, 1890, 0, 0], [-348, 0, 0, 0], [0, 0, -480, 0], [900, 0, 0, 0], [0, 0, 0, -285], [0, -2215, 0, 0], [0, 0, 75, 0], [0, 0, 0, 570], [0, 0, 510, 0], [0, -2045, 0, 0], [2436, 0, 0, 0], [0, 0, 0, 320], [0, 0, -324, 0], [0, 0, -84, 0], [0, 0, 0, 210], [0, 0, -510, 0], [2507, 0, 0, 0], [0, 1050, 0, 0], [-2334, 0, 0, 0], [0, 0, -105, 0], [0, 0, 0, 436], [0, -1200, 0, 0], [-610, 0, 0, 0], [0, -1435, 0, 0], [-1410, 0, 0, 0], [0, 0, 225, 0], [0, 0, -79, 0], [0, 0, 0, 504], [0, 0, 0, -380], [0, 882, 0, 0], [0, 0, 0, 240], [0, 0, 0, 544], [0, 90, 0, 0], [0, 0, -175, 0], [0, 0, 0, 611], [2930, 0, 0, 0], [0, 2971, 0, 0], [1450, 0, 0, 0], [0, -3125, 0, 0], [-390, 0, 0, 0], [0, 0, -260, 0], [-1834, 0, 0, 0], [-216, 0, 0, 0], [0, 0, -174, 0], [0, 1666, 0, 0], [0, 0, -204, 0], [0, 0, 435, 0], [0, 0, 0, 39], [0, 0, 630, 0], [-875, 0, 0, 0], [0, -1380, 0, 0], [0, 761, 0, 0], [0, 0, 0, -90], [0, -1854, 0, 0], [1320, 0, 0, 0], [0, 0, 0, -36], [-1488, 0, 0, 0], [0, 0, -120, 0], [0, 1152, 0, 0], [0, -3127, 0, 0], [0, 2130, 0, 0], [-876, 0, 0, 0], [0, 0, 0, 295], [1225, 0, 0, 0], [0, 660, 0, 0], [0, -3746, 0, 0]]
