
# q-expansion of newform 8048.2.a.f, 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 = 8048
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], [0], [0], [-4], [2], [-2], [2], [-8], [0], [-8], [8], [6], [-4], [8], [12], [4], [2], [4], [-4], [-6], [-8], [-12], [18], [-6], [16], [20], [-2], [16], [18], [-12], [-8], [-18], [2], [-12], [8], [4], [22], [12], [-18], [-26], [-12], [8], [6], [-2], [8], [22], [16], [10], [22], [-18], [16], [30], [-6], [22], [-24], [-12], [24], [-8], [-2], [-8], [-22], [10], [12], [-26], [-10], [6], [34], [10], [4], [18], [-12], [32], [-38], [12], [24], [38], [18], [-6], [22], [-14], [26], [-24], [-18], [0], [-12], [-18], [22], [-32], [-8], [-30], [-20], [-20], [-18], [14], [-1], [2], [22], [6], [20], [36], [-30], [34], [-10], [-18], [-22], [28], [26], [0], [38], [8], [4], [2], [26], [-40], [-2], [-2], [-48], [-6], [48], [42], [14], [24], [34], [44], [-30], [12], [24], [-40], [4], [-32], [-32], [-20], [-52], [26], [30], [28], [50], [-22], [-26], [16], [-24], [-4], [-56], [0], [-8], [-38], [-26], [6], [24], [52], [-2], [8], [48], [38], [36], [16], [6], [34], [42], [50], [-10], [-28], [-40], [2], [32], [48], [-48], [-14], [34], [-48], [-16], [0], [26], [-16], [62], [-22], [-24], [16], [46], [16], [40], [-4], [-6], [32], [-36], [-16], [32], [46], [0], [2], [-50], [-10], [-62], [-50], [10], [34], [10], [-38], [4], [10], [32], [-14], [-34], [24], [-34], [16], [-18], [30], [-40], [-2], [-52], [-16], [28], [-48], [-10], [-60], [2], [8], [14], [32], [56], [-50], [-12], [16], [-18], [14], [16], [-8], [-48], [-32], [-58], [-56], [-54], [14], [60], [30], [-60], [-24], [-32], [-48], [28], [16], [12], [-62], [32], [8], [66], [70], [-36], [-2], [14], [-32], [-58], [-26], [-14], [30], [-28], [-22], [30], [-16], [48], [4], [10], [18], [16], [-18], [-14], [-74], [18], [44], [2], [30], [-16], [-10], [24], [-46], [32], [-38], [66], [-28], [16], [72], [-18], [-42], [4], [-18], [-6], [-40], [-30], [-26], [-28], [18], [-54], [-52], [64], [-76], [-48], [-2], [70], [82], [-10], [36], [-18], [50], [-58], [-30], [-72], [48], [-36], [24], [72], [-70], [22], [-32], [58], [54], [-56], [18], [-14], [-58], [-2], [6], [64], [82], [-50], [46], [22], [36], [34], [-40], [74], [28], [-80], [70], [56], [-26], [90], [-22], [8], [26], [-38], [-2], [-64], [-80], [30], [-74], [18], [16], [-12], [-2], [14], [-12], [8], [-60], [-6], [48], [-18], [-2], [36], [36], [42], [-56], [-78], [-6], [-54], [78], [-8], [38], [-20], [20], [24], [76], [-88], [-66], [68], [60], [30], [30], [14], [-36], [6], [-56], [90], [92], [92], [16], [-70], [20], [96], [-22], [-72], [-24], [16], [-96], [-14], [76], [-18], [-20], [54], [30], [-42], [48], [38], [28], [8], [50], [90], [16], [46], [26], [-18], [58], [-20], [82], [-30], [40], [-76], [46], [8], [30], [46], [-48], [26], [6], [84], [-44], [-14], [46], [56]]
