
# q-expansion of newform 4032.2.a.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
    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 = 4032
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], [-2], [-1], [-4], [6], [2], [-4], [4], [-2], [8], [10], [2], [-8], [0], [-10], [-12], [-10], [8], [-12], [2], [0], [12], [-6], [2], [-10], [-8], [-12], [2], [14], [8], [-20], [-18], [-20], [6], [-16], [-2], [-16], [0], [6], [-20], [-2], [-12], [-14], [6], [16], [16], [-8], [20], [-18], [14], [-12], [-30], [-4], [18], [12], [-2], [-32], [18], [30], [12], [6], [-4], [-8], [-14], [-2], [8], [-14], [-28], [-18], [-6], [12], [-24], [34], [8], [8], [38], [-34], [-10], [-22], [12], [34], [-36], [-6], [8], [28], [-18], [-6], [-26], [16], [36], [0], [-32], [-36], [16], [0], [-18], [-38], [-28], [26], [32], [30], [-20], [-18], [-16], [2], [12], [42], [36], [10], [8], [10], [-18], [20], [0], [-26], [-28], [16], [14], [44], [-18], [50], [-18], [28], [-12], [30], [-38], [-8], [-40], [14], [-32], [28], [16], [26], [-30], [26], [-42], [4], [-42], [-10], [4], [-10], [32], [4], [14], [16], [-42], [18], [-12], [-52], [-14], [42], [-16], [16], [-48], [12], [8], [-22], [10], [6], [44], [54], [8], [28], [22], [-24], [8], [62], [2], [30], [36], [30], [20], [-38], [32], [2], [-40], [6], [32], [6], [8], [-12], [-38], [2], [20], [30], [10], [4], [-6], [8], [-22], [-12], [0], [-42], [-44], [-14], [-14], [2], [2], [56], [30], [0], [-10], [10], [12], [-66], [64], [-36], [6], [36], [-62], [-2], [16], [12], [40], [-46], [56], [-46], [-20], [-34], [-6], [32], [-66], [-16], [28], [26], [-6], [-12], [-32], [-60], [-14], [-28], [-24], [54], [-4], [0], [34], [-26], [12], [-16], [-28], [-60], [-72], [26], [18], [-48], [-48], [52], [16], [-36], [-22], [-38], [-28], [-62], [-18], [-12], [26], [-28], [6], [-14], [-40], [-12], [-10], [22], [-78], [28], [-34], [-30], [8], [38], [14], [72], [34], [80], [26], [-64], [12], [26], [-6], [-28], [48], [-80], [-56], [-2], [52], [44], [34], [6], [40], [2], [-66], [60], [-34], [-60], [26], [-74], [24], [-58], [-20], [-36], [-46], [-18], [32], [60], [32], [-14], [36], [30], [76], [-46], [0], [54], [54], [40], [-4], [-30], [-12], [-20], [-6], [-10], [36], [26], [-50], [24], [54], [-14], [-16], [28], [12], [38], [10], [62], [40], [-84], [-56], [-28], [-14], [-30], [66], [-8], [-78], [-66], [-42], [8], [30], [60], [-34], [88], [-32], [54], [-60], [58], [-50], [0], [-46], [2], [24], [44], [30], [44], [-14], [-70], [-4], [-60], [-12], [90], [46], [-56], [-70], [-84], [20], [-60], [22], [80], [-38], [76], [36], [-14], [82], [42], [70], [-74], [-72], [-18], [-68], [-48], [-56], [22], [-40], [48], [18], [62], [-12], [68], [60], [26], [0], [42], [8], [-42], [-26], [-18], [64], [-78], [-74], [0], [-38], [14], [-20], [-20], [34], [-10], [-44], [80], [-38], [-66], [12], [8], [-62], [24], [-2], [78], [36], [-12], [82], [94], [4], [-18], [12], [16]]
