
# q-expansion of newform 7600.2.a.f, 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 = 7600
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], [-2], [0], [4], [-4], [4], [2], [1], [0], [2], [-8], [-4], [6], [0], [12], [8], [0], [2], [-14], [-8], [-6], [4], [4], [14], [12], [10], [14], [-18], [-2], [12], [-6], [-12], [-14], [-12], [10], [12], [14], [-24], [-2], [16], [-24], [-26], [8], [-4], [10], [8], [-8], [22], [22], [-6], [6], [24], [-10], [20], [24], [16], [-10], [8], [-2], [-10], [20], [-24], [-26], [32], [14], [4], [8], [-28], [20], [-26], [-30], [32], [-12], [24], [-4], [2], [6], [22], [10], [34], [12], [-22], [-28], [12], [12], [12], [10], [22], [14], [40], [36], [0], [-2], [20], [36], [-16], [34], [18], [26], [26], [22], [-42], [-14], [-6], [-28], [30], [20], [-10], [20], [6], [-42], [38], [2], [20], [24], [-10], [20], [0], [14], [8], [26], [8], [24], [-14], [-36], [2], [-34], [48], [28], [42], [-28], [-22], [-44], [2], [-42], [-50], [36], [14], [0], [-6], [-28], [-18], [44], [-6], [38], [12], [58], [-32], [4], [26], [8], [30], [-8], [10], [46], [-12], [-32], [-30], [-14], [-42], [-36], [-32], [-48], [-20], [32], [-22], [-4], [-34], [30], [-50], [-12], [6], [-40], [-58], [-16], [-26], [-4], [42], [-30], [58], [-16], [4], [-4], [-24], [8], [30], [-32], [10], [18], [24], [-56], [-20], [-52], [-30], [48], [-12], [10], [18], [38], [12], [10], [28], [-28], [62], [52], [-18], [24], [-10], [-30], [-12], [6], [-22], [-16], [42], [0], [-42], [-52], [-30], [-30], [-6], [2], [56], [-2], [64], [-42], [-10], [24], [24], [-22], [-4], [2], [-44], [28], [-42], [44], [76], [30], [18], [12], [-68], [-54], [-68], [-68], [10], [-60], [24], [12], [0], [44], [-12], [-22], [-62], [-24], [34], [42], [4], [-26], [10], [-32], [34], [-74], [-6], [-6], [-32], [62], [8], [-22], [-2], [46], [-22], [-58], [50], [18], [40], [68], [-52], [32], [18], [-74], [-76], [18], [-32], [72], [-2], [20], [40], [-26], [-52], [8], [-6], [-58], [-72], [-12], [72], [80], [42], [48], [-46], [-4], [0], [26], [48], [-48], [-74], [-76], [-20], [-30], [-46], [-8], [-30], [-4], [10], [66], [26], [48], [62], [4], [-12], [14], [66], [-20], [26], [-6], [2], [-86], [30], [24], [66], [2], [78], [66], [0], [-16], [32], [52], [24], [-82], [0], [42], [8], [-24], [-6], [14], [48], [32], [-16], [-10], [-86], [16], [38], [44], [-48], [-30], [-62], [10], [-48], [8], [-12], [42], [-30], [30], [-90], [30], [-28], [-28], [-36], [6], [74], [70], [28], [40], [12], [-66], [-40], [38], [12], [-16], [-50], [-14], [28], [-42], [14], [28], [30], [0], [-58], [96], [38], [16], [-42], [-94], [-56], [76], [12], [12], [0], [-52], [22], [-4], [-10], [62], [38], [-82], [-48], [-90], [-8], [-98], [2], [98], [4], [64], [-94], [46], [-12], [-86], [-18], [60], [-102], [-86], [-22], [70], [96], [76], [-92], [-76], [-64], [-78], [78], [44], [-40]]
