
# q-expansion of newform 1728.2.a.m, downloaded from the LMFDB on 20 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 = 1728
weight = 2
poly_data = [0, 1]

# The basis for the coefficient ring is just the power basis
# in the root of the defining polynomial above.
hecke_ring_character_values = None
aps_data = [[0], [0], [0], [-5], [0], [7], [0], [-1], [0], [0], [4], [1], [0], [8], [0], [0], [0], [13], [11], [0], [17], [13], [0], [0], [5], [0], [7], [0], [-2], [0], [-20], [0], [0], [-7], [0], [-23], [-14], [17], [0], [0], [0], [19], [0], [-25], [0], [-17], [29], [28], [0], [22], [0], [0], [-31], [0], [0], [0], [0], [1], [-26], [0], [32], [0], [-16], [0], [-13], [0], [-31], [29], [0], [-23], [0], [0], [31], [25], [-37], [0], [0], [34], [0], [-7], [0], [-41], [0], [2], [28], [0], [0], [-10], [0], [43], [0], [0], [19], [0], [32], [0], [0], [0], [35], [-17], [41], [0], [0], [0], [47], [35], [0], [0], [0], [26], [-29], [37], [0], [-49], [1], [0], [-40], [0], [0], [0], [-11], [-13], [0], [0], [8], [0], [31], [0], [-44], [-50], [-16], [0], [-11], [55], [0], [47], [0], [-25], [0], [0], [56], [0], [-47], [0], [-53], [0], [-23], [0], [-43], [0], [25], [0], [-55], [0], [59], [0], [52], [0], [35], [0], [0], [0], [61], [0], [0], [0], [-17], [10], [-19], [0], [0], [-14], [0], [5], [7], [0], [-64], [0], [37], [-62], [-59], [0], [22], [0], [0], [0], [19], [-67], [29], [0], [-55], [0], [53], [0], [0], [0], [-61], [67], [0], [0], [0], [49], [-29], [-67], [0], [0], [-71], [0], [0], [11], [71], [0], [13], [0], [0], [-49], [4], [0], [0], [0], [43], [-68], [0], [-20], [0], [13], [0], [0], [-41], [0], [-65], [56], [76], [0], [77], [0], [-43], [0], [0], [0], [0], [71], [25], [73], [0], [0], [-35], [0], [32], [0], [-50], [0], [0], [77], [0], [0], [-53], [80], [0], [-70], [-5], [0], [7], [-35], [0], [-64], [0], [0], [-40], [0], [-83], [-19], [-10], [-83], [14], [55], [0], [82], [74], [0], [0], [-68], [0], [49], [29], [0], [17], [0], [-77], [0], [0], [0], [0], [0], [-62], [0], [-77], [0], [0], [-49], [83], [0], [52], [0], [29], [-34], [0], [79], [0], [73], [0], [0], [0], [65], [0], [38], [0], [0], [-82], [0], [59], [11], [0], [-92], [0], [-91], [-88], [8], [0], [0], [-41], [0], [-23], [0], [-16], [0], [1], [0], [86], [-20], [-95], [0], [0], [13], [0], [0], [-74], [-64], [0], [0], [-97], [89], [0], [28], [97], [0], [0], [0], [0], [0], [85], [0], [0], [0], [-91], [95], [0], [-53], [71], [0], [-91], [0], [0], [-101], [55], [0], [0], [-73], [0], [5], [0], [0], [-71], [0], [47], [0], [-44], [-101], [17], [0], [62], [0], [0], [-49], [0], [65], [73], [0], [104], [0], [-14], [0], [76], [0], [0], [-92], [-5], [0], [-7], [0], [98], [0], [0], [-31], [65], [0], [0], [-95], [0], [0], [0], [106], [0], [0], [-70], [0], [0], [0], [56], [0]]
