
# q-expansion of newform 7245.2.a.m, 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 = 7245
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], [-1], [1], [2], [-4], [2], [0], [-1], [6], [-2], [4], [2], [-8], [8], [0], [-2], [-14], [-12], [0], [8], [10], [-12], [-6], [-14], [-18], [8], [-4], [2], [8], [-16], [6], [20], [-10], [-2], [-20], [10], [-20], [16], [4], [-4], [-26], [18], [22], [-18], [-20], [8], [16], [-4], [-14], [-6], [-8], [-6], [-16], [0], [-20], [-10], [-10], [10], [18], [4], [10], [-20], [6], [-26], [-10], [-4], [0], [-36], [-26], [4], [-10], [32], [28], [26], [-24], [6], [0], [14], [34], [12], [-10], [-10], [2], [-14], [-4], [10], [-20], [-18], [-8], [-20], [-28], [-8], [12], [-24], [0], [-6], [2], [44], [2], [28], [-40], [-4], [-18], [10], [-8], [28], [-44], [12], [6], [36], [16], [8], [24], [10], [18], [12], [28], [-18], [30], [-26], [-10], [14], [-44], [14], [-6], [6], [-30], [-40], [-42], [0], [48], [2], [24], [6], [10], [-42], [12], [2], [30], [30], [42], [-32], [36], [-30], [4], [16], [-20], [14], [48], [46], [-18], [36], [28], [-24], [38], [38], [6], [-14], [30], [28], [44], [-8], [-56], [-48], [-16], [4], [12], [-26], [-24], [-30], [26], [-30], [-52], [30], [30], [40], [2], [-24], [34], [56], [-40], [14], [0], [-24], [-2], [38], [-4], [-6], [-50], [-8], [60], [-30], [42], [36], [-14], [-50], [24], [22], [-56], [50], [-62], [-12], [-14], [-48], [-12], [-52], [44], [54], [-38], [-2], [42], [-24], [-60], [30], [-26], [24], [-50], [20], [70], [38], [40], [66], [20], [24], [-22], [-54], [8], [56], [-24], [-18], [-52], [-6], [-18], [60], [32], [-14], [-28], [32], [-34], [0], [2], [-8], [-38], [-12], [-38], [40], [40], [-70], [-20], [-28], [18], [-12], [46], [44], [-48], [10], [12], [48], [-32], [8], [-60], [-18], [-14], [-12], [4], [-54], [-54], [-36], [66], [10], [-20], [-2], [-58], [-68], [36], [-28], [10], [6], [-52], [-56], [-10], [-16], [-6], [-72], [24], [56], [-32], [-46], [54], [26], [60], [-66], [-64], [42], [74], [-32], [36], [-62], [-56], [-38], [36], [58], [-28], [-54], [38], [52], [34], [-54], [-74], [-84], [18], [14], [-52], [24], [2], [2], [-60], [62], [82], [-84], [-32], [50], [48], [32], [42], [30], [-52], [60], [-40], [2], [-2], [-30], [-24], [-26], [0], [-22], [42], [62], [72], [22], [76], [22], [54], [64], [44], [-50], [-52], [-84], [50], [-70], [-18], [-18], [32], [86], [36], [-36], [-84], [74], [-40], [72], [6], [-72], [78], [4], [26], [72], [-4], [-34], [-32], [-30], [48], [-2], [80], [-50], [-38], [-66], [-94], [-54], [-28], [-30], [72], [-32], [54], [-16], [84], [8], [32], [88], [-24], [-70], [-52], [60], [-80], [-18], [28], [60], [22], [38], [58], [50], [10], [8], [-58], [14], [18], [-20], [66], [52], [-6], [-56], [6], [-36], [-94], [24], [-2], [88], [16], [30], [-8], [14], [78], [48], [-30], [-48], [24], [64], [-54], [-38], [66]]
