
# q-expansion of newform 1200.2.a.m, 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
    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 = 1200
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], [1], [0], [-2], [-2], [6], [2], [0], [4], [0], [8], [2], [2], [4], [8], [6], [-10], [2], [8], [-12], [-4], [0], [4], [-10], [-8], [-8], [14], [-12], [10], [6], [-2], [18], [-18], [20], [-20], [8], [22], [-16], [-12], [-14], [-10], [2], [-12], [-4], [22], [0], [-12], [-26], [28], [-10], [-14], [-20], [22], [18], [-18], [4], [0], [8], [2], [-18], [-16], [6], [-12], [-12], [-4], [2], [8], [-28], [-12], [10], [-14], [0], [-2], [6], [-20], [-16], [20], [2], [22], [-10], [10], [22], [-32], [-4], [0], [-36], [30], [32], [12], [-6], [-12], [-20], [18], [18], [0], [24], [0], [22], [-16], [-38], [28], [-18], [44], [-10], [8], [32], [-12], [6], [0], [2], [-22], [26], [2], [-20], [-32], [2], [24], [48], [26], [50], [2], [36], [2], [4], [8], [32], [30], [40], [18], [-14], [-40], [24], [-32], [2], [-18], [-30], [-54], [-32], [2], [-30], [-52], [-8], [-6], [28], [-30], [-40], [-14], [42], [-20], [-36], [22], [-18], [24], [-12], [-12], [48], [0], [30], [-8], [-28], [-12], [46], [38], [18], [42], [-16], [8], [-18], [-10], [-54], [30], [-18], [-32], [-44], [-40], [30], [28], [12], [34], [-10], [-62], [-22], [-14], [-18], [-16], [-60], [-58], [-16], [50], [-52], [36], [4], [-52], [12], [-12], [6], [-18], [6], [-18], [-16], [60], [48], [-18], [-50], [-50], [-18], [0], [4], [-50], [8], [52], [12], [14], [-12], [20], [22], [38], [22], [8], [-34], [-18], [-40], [-30], [-46], [-12], [30], [-34], [-40], [-22], [-2], [46], [20], [-32], [-18], [-56], [28], [30], [46], [30], [-12], [-36], [-32], [34], [10], [26], [-60], [-42], [-42], [40], [4], [42], [2], [68], [70], [46], [-30], [-38], [8], [22], [-68], [54], [28], [70], [-14], [-18], [40], [-60], [2], [64], [66], [62], [68], [76], [-40], [-28], [14], [-52], [-70], [-58], [18], [-36], [-32], [8], [22], [-52], [28], [-24], [82], [40], [50], [-28], [-12], [-54], [-42], [26], [60], [48], [-54], [-50], [-12], [76], [42], [40], [-36], [-32], [32], [68], [-10], [20], [6], [-36], [0], [-18], [-36], [-32], [-10], [-10], [68], [-4], [-30], [88], [-68], [72], [-86], [-54], [-58], [-60], [-36], [-32], [26], [22], [62], [40], [44], [-12], [-12], [-50], [66], [-38], [-2], [6], [-38], [-40], [-32], [-34], [30], [-18], [-52], [68], [42], [28], [-48], [-48], [54], [50], [-54], [0], [-42], [-58], [4], [62], [-78], [48], [-50], [-72], [-64], [-18], [-66], [42], [-82], [-20], [64], [0], [-72], [-78], [-50], [8], [-24], [30], [-8], [12], [-74], [38], [82], [60], [-36], [-72], [-38], [84], [-92], [50], [6], [30], [8], [8], [-64], [40], [10], [88], [12], [-50], [-34], [38], [-18], [0], [-32], [42], [102], [-16], [90], [-104], [-18], [-36], [8], [92], [32], [40], [78], [-58], [-36], [60], [22], [88], [-30], [-44], [-18], [-76], [10], [48], [40]]
