
# q-expansion of newform 7350.2.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 = 7350
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], [-1], [0], [0], [0], [2], [2], [-2], [8], [-8], [-4], [-6], [-10], [2], [6], [6], [-12], [-2], [-14], [6], [10], [4], [-12], [14], [-14], [6], [4], [0], [-2], [10], [8], [-8], [-2], [-22], [20], [20], [-2], [2], [2], [-12], [-12], [-2], [22], [-20], [-6], [-16], [-4], [-20], [16], [10], [-14], [18], [-24], [-24], [30], [0], [-30], [20], [-2], [18], [20], [-24], [12], [-8], [6], [34], [28], [-12], [28], [-30], [6], [-14], [16], [-22], [12], [-26], [-4], [-38], [10], [-24], [-12], [-18], [-10], [-14], [36], [-4], [-26], [-32], [14], [-4], [-36], [-24], [-32], [-4], [-12], [26], [-14], [-18], [-4], [-30], [26], [18], [24], [10], [4], [-26], [0], [-14], [-34], [-8], [-12], [10], [-6], [22], [-24], [-42], [-36], [14], [14], [16], [22], [-20], [-12], [24], [-34], [-8], [26], [-8], [36], [2], [-4], [-32], [-24], [34], [34], [20], [-24], [-28], [-20], [18], [-22], [0], [-4], [-12], [-22], [-24], [10], [-10], [-42], [32], [-42], [18], [10], [-30], [-38], [22], [-16], [-22], [-42], [-14], [24], [34], [32], [0], [34], [-18], [16], [-6], [-26], [-28], [-40], [-42], [10], [-36], [-4], [-30], [20], [4], [-44], [50], [-28], [52], [58], [-14], [0], [46], [-38], [44], [-50], [24], [34], [24], [36], [-30], [32], [26], [-10], [-58], [22], [30], [-28], [40], [-2], [68], [40], [-8], [-68], [-12], [-50], [-18], [12], [22], [56], [-28], [64], [-52], [32], [30], [32], [-34], [-22], [24], [-46], [-28], [24], [26], [62], [-42], [24], [-68], [30], [22], [8], [46], [44], [66], [-8], [-62], [-60], [-8], [-48], [62], [-16], [26], [-22], [-24], [76], [0], [-52], [24], [54], [14], [-32], [20], [0], [60], [46], [76], [-48], [26], [-4], [4], [62], [6], [-66], [38], [12], [-18], [-50], [-46], [22], [-2], [-74], [-4], [-34], [32], [64], [-22], [50], [-12], [70], [20], [18], [50], [68], [-62], [-16], [30], [28], [26], [36], [48], [26], [0], [-18], [42], [-20], [-40], [48], [-4], [70], [-82], [88], [-24], [-12], [8], [-16], [14], [14], [26], [22], [-48], [82], [10], [-8], [-68], [28], [-62], [-14], [-66], [-50], [-68], [-18], [-56], [62], [48], [36], [-12], [-24], [-34], [-10], [-66], [-72], [-44], [-12], [68], [50], [2], [28], [-56], [82], [2], [70], [72], [66], [-4], [-30], [-10], [-72], [56], [66], [16], [-48], [-64], [-86], [-42], [72], [44], [-62], [24], [34], [-78], [72], [-16], [-28], [8], [56], [-52], [-26], [-72], [-66], [24], [16], [44], [62], [-24], [-30], [34], [-26], [-26], [42], [-62], [0], [22], [50], [54], [-12], [-78], [-14], [-38], [82], [-36], [0], [44], [-90], [-64], [0], [-30], [-44], [84], [10], [-78], [92], [18], [-38], [-8], [22], [-34], [28], [60], [62], [42], [84], [92], [-60], [46], [-78], [-8], [-2], [66], [-12], [-14], [88], [-40], [70], [-80], [-60], [-26], [2], [-64], [64], [-96], [-20], [22], [-58], [22], [-26], [-34], [22], [-76], [-24], [-6], [106], [26], [-52], [-90], [68], [-50], [84], [-66], [110], [-48], [60], [-90], [4], [-36], [38], [-16], [22], [10], [28], [4], [-24], [-10], [-68], [4], [-76], [-44], [-50], [10], [-12], [12], [-48], [66]]
