
# q-expansion of newform 4788.2.a.f, downloaded from the LMFDB on 06 September 2025.

# 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 = 4788
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], [2], [1], [2], [-6], [-2], [1], [-6], [8], [-8], [-10], [-8], [-8], [-2], [8], [0], [-6], [-4], [0], [6], [0], [-14], [16], [-2], [-18], [-8], [12], [-6], [12], [16], [6], [-6], [4], [2], [8], [-14], [4], [8], [8], [4], [26], [14], [2], [-18], [12], [-4], [-8], [12], [-2], [-18], [22], [22], [2], [-28], [-10], [-24], [-28], [-26], [-8], [-16], [-24], [28], [22], [22], [-12], [28], [-34], [-14], [-2], [6], [6], [4], [6], [36], [4], [-26], [14], [-12], [-34], [-10], [-14], [16], [-6], [-40], [18], [-24], [22], [-26], [-28], [-6], [2], [-24], [18], [-12], [2], [24], [-8], [-4], [2], [28], [-6], [20], [32], [-24], [-2], [-38], [26], [20], [-38], [40], [34], [-22], [4], [-40], [12], [-32], [-18], [42], [-32], [46], [-46], [28], [44], [8], [42], [-6], [-18], [-4], [-30], [20], [8], [-48], [-18], [30], [-2], [24], [-4], [-24], [30], [20], [42], [8], [-36], [26], [-36], [-42], [32], [-28], [24], [22], [-42], [-20], [24], [4], [-12], [-48], [-46], [10], [-16], [50], [-24], [16], [20], [28], [32], [32], [26], [30], [-34], [-4], [-2], [42], [26], [-40], [42], [52], [42], [-8], [14], [44], [44], [-6], [-12], [42], [54], [-46], [-12], [-34], [-42], [14], [10], [-20], [12], [18], [-48], [-30], [-30], [-6], [-6], [0], [40], [2], [-66], [-18], [-58], [-52], [52], [42], [-20], [22], [50], [20], [16], [-12], [-38], [52], [-24], [-24], [-50], [58], [-64], [-20], [24], [-12], [-6], [36], [-16], [32], [-30], [-26], [-60], [-32], [60], [-4], [-18], [-2], [-42], [26], [-68], [4], [-40], [52], [46], [-60], [66], [56], [-36], [20], [-62], [-10], [18], [-30], [6], [-62], [74], [10], [-4], [12], [-18], [-16], [-72], [10], [-74], [-38], [20], [-32], [2], [4], [6], [-58], [60], [42], [32], [-50], [56], [38], [-82], [34], [10], [4], [-32], [26], [-58], [8], [-30], [34], [-52], [-32], [20], [18], [-70], [-16], [48], [30], [10], [8], [-54], [60], [-4], [-34], [-32], [76], [-84], [12], [38], [72], [26], [-46], [22], [-78], [30], [-36], [-4], [74], [46], [-10], [-60], [-18], [-6], [20], [74], [-52], [-24], [42], [-42], [44], [36], [-16], [26], [38], [-24], [-32], [22], [-32], [58], [-10], [-56], [18], [-60], [-26], [-30], [12], [-56], [24], [88], [-10], [4], [48], [38], [52], [-90], [-50], [-48], [-18], [40], [-22], [42], [-62], [-32], [-26], [30], [-56], [-36], [-4], [-6], [-90], [-64], [10], [-2], [12], [-54], [-72], [-52], [-30], [60], [50], [-14], [-14], [30], [60], [70], [-60], [-66], [-84], [-4], [-84], [30], [-60], [-48], [6], [20], [90], [44], [16], [14], [-64], [44], [28], [-18], [-50], [-66], [64], [-60], [-80], [-84], [2], [-16], [60], [94], [-22], [54], [40], [8], [86], [-42], [-28], [24], [46], [84], [0], [-66], [34], [4], [2], [0], [108], [6], [-76], [-66]]
