
# q-expansion of newform 2925.2.a.m, downloaded from the LMFDB on 19 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 = 2925
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 = [[1], [0], [0], [-2], [-4], [1], [4], [6], [0], [-4], [-10], [2], [-6], [8], [8], [4], [12], [2], [10], [0], [6], [12], [4], [14], [14], [12], [4], [-4], [-2], [-16], [8], [-16], [-10], [12], [-18], [10], [-18], [-6], [24], [-24], [0], [-10], [-12], [26], [-18], [8], [0], [14], [20], [-10], [8], [0], [22], [12], [-8], [4], [12], [14], [-10], [-30], [4], [2], [22], [-4], [-2], [-18], [18], [-14], [-32], [-6], [6], [-16], [8], [-22], [-2], [24], [8], [18], [22], [30], [-16], [22], [-40], [2], [-36], [-20], [30], [22], [-14], [-18], [-8], [-24], [22], [36], [-38], [-12], [42], [12], [12], [14], [36], [26], [4], [28], [-12], [-38], [20], [-14], [8], [10], [8], [-6], [-18], [22], [-14], [-40], [-14], [28], [0], [36], [26], [18], [36], [-4], [34], [40], [-50], [-20], [24], [-34], [30], [-24], [20], [34], [18], [2], [50], [22], [12], [28], [30], [-10], [4], [20], [-2], [-40], [-30], [48], [16], [48], [42], [-28], [4], [56], [-44], [12], [44], [10], [-54], [6], [-12], [48], [-58], [-52], [-54], [16], [-48], [58], [-10], [-40], [-44], [-46], [-28], [-34], [28], [16], [-26], [-10], [-12], [-10], [-10], [40], [34], [2], [24], [-24], [42], [46], [46], [-64], [-34], [-36], [20], [-18], [36], [56], [-22], [-2], [-6], [-24], [2], [40], [-34], [-46], [-60], [-44], [-34], [-52], [-18], [-52], [-30], [-8], [-4], [-12], [0], [-14], [-20], [-64], [-56], [50], [10], [46], [38], [46], [-32], [62], [-4], [-48], [-72], [4], [58], [28], [2], [-24], [-4], [0], [70], [2], [48], [-60], [20], [-28], [36], [34], [-6], [-20], [38], [4], [30], [48], [38], [-46], [0], [-74], [0], [-12], [-30], [6], [44], [66], [-24], [0], [-42], [34], [66], [8], [26], [-54], [-18], [48], [-46], [-70], [62], [-4], [-22], [42], [-4], [-14], [18], [-36], [-8], [-66], [-24], [-54], [14], [56], [-66], [-38], [18], [-52], [24], [56], [-70], [-12], [-50], [40], [-48], [28], [60], [46], [-34], [62], [28], [-24], [20], [78], [-60], [-14], [-8], [-70], [36], [42], [60], [-64], [-48], [22], [28], [0], [-66], [0], [40], [-78], [48], [58], [82], [-6], [42], [34], [12], [-56], [22], [72], [-12], [36], [30], [36], [-14], [14], [-2], [-52], [-46], [24], [30], [20], [-66], [64], [-78], [58], [0], [20], [-22], [-2], [62], [-8], [-86], [-72], [80], [12], [24], [24], [-78], [48], [-4], [60], [72], [-86], [14], [-30], [46], [80], [16], [24], [16], [-44], [30], [-84], [40], [-46], [-24], [70], [-18], [38], [-26], [42], [26], [16], [10], [-6], [-10], [12], [-58], [74], [-36], [88], [24], [-74], [10], [0], [-40], [-78], [-14], [-92], [6], [18], [18], [-92], [-34], [-18], [6], [-36], [6], [-36], [0], [8], [34], [-72], [-48], [-28], [-46], [-8], [12], [-38], [104], [8], [-46], [-102], [32], [62], [-70], [-52]]
