
# q-expansion of newform 1800.2.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 = 1800
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], [0], [-4], [2], [2], [-4], [-8], [-6], [8], [-6], [6], [-4], [0], [-2], [-4], [-2], [4], [-8], [-10], [-8], [-4], [6], [-2], [18], [-16], [-12], [-2], [18], [8], [4], [-6], [-12], [-14], [-16], [2], [-12], [24], [6], [-12], [6], [0], [-2], [-18], [16], [-20], [8], [12], [22], [10], [16], [18], [-20], [2], [-8], [10], [8], [26], [-26], [28], [-18], [-12], [24], [6], [6], [20], [-18], [-12], [30], [2], [24], [8], [10], [20], [0], [2], [-14], [30], [-6], [-12], [-10], [-32], [14], [0], [20], [14], [22], [26], [-8], [-36], [16], [32], [12], [12], [24], [-6], [-26], [-4], [-18], [-44], [-26], [28], [-10], [36], [-2], [-44], [-14], [-24], [-38], [40], [-38], [42], [-44], [16], [14], [-12], [8], [6], [-12], [-10], [-34], [-2], [4], [-4], [-6], [-10], [32], [-48], [-14], [-4], [-8], [24], [-38], [22], [2], [-18], [-28], [22], [-26], [4], [-30], [16], [-28], [-50], [24], [10], [42], [-12], [-32], [18], [-50], [4], [8], [-4], [-16], [16], [-50], [-42], [-6], [12], [-54], [16], [-36], [-30], [-24], [40], [26], [-46], [-50], [-36], [30], [40], [-10], [-40], [-26], [-28], [-30], [-16], [14], [-8], [52], [-54], [-54], [48], [18], [50], [36], [10], [0], [62], [-12], [-20], [-22], [-36], [-22], [-14], [-46], [2], [24], [-38], [8], [-38], [34], [60], [6], [56], [12], [22], [4], [-18], [18], [32], [-28], [-8], [-22], [40], [62], [-24], [54], [-10], [0], [-18], [-8], [60], [-10], [-6], [-48], [-48], [-52], [66], [28], [-40], [-10], [60], [-64], [50], [-50], [-20], [-40], [-36], [-28], [-64], [46], [18], [40], [56], [-60], [52], [-16], [-78], [30], [8], [10], [54], [-12], [38], [-4], [78], [-58], [8], [60], [22], [66], [-14], [-4], [-38], [-26], [76], [62], [-50], [52], [38], [40], [14], [64], [-12], [62], [10], [-76], [-80], [16], [-8], [-58], [28], [-16], [-50], [-18], [-16], [-2], [26], [12], [-38], [12], [18], [74], [-40], [-66], [12], [-44], [-74], [-58], [-40], [-20], [84], [-2], [36], [-34], [-56], [58], [48], [-30], [46], [20], [-8], [-22], [84], [-48], [-2], [78], [-20], [38], [58], [56], [58], [18], [12], [-84], [-32], [30], [-2], [-42], [24], [-36], [-12], [84], [-2], [66], [-22], [-72], [-22], [-54], [-78], [-48], [-26], [20], [-42], [-52], [0], [-34], [-84], [22], [-22], [24], [-10], [26], [-64], [76], [34], [-24], [-38], [-42], [48], [-20], [68], [-42], [-42], [32], [-38], [52], [36], [80], [2], [16], [82], [-60], [96], [-34], [-66], [-26], [42], [-70], [-48], [-30], [-20], [24], [24], [58], [-4], [-48], [-62], [-98], [12], [-60], [-72], [-58], [24], [6], [-28], [-46], [62], [18], [-40], [-22], [-46], [64], [2], [-66], [100], [-12], [46], [46], [-76], [76], [-42], [-22], [0], [-32], [-30], [-40], [10], [58], [-16], [76], [-10], [-10], [28], [6], [-12], [-88]]
