
# q-expansion of newform 7803.2.a.m, downloaded from the LMFDB on 03 May 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 = 7803
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], [5], [0], [-7], [0], [8], [0], [0], [-7], [-10], [0], [5], [0], [0], [0], [-13], [-16], [0], [-7], [17], [0], [0], [5], [0], [-7], [0], [2], [0], [-1], [0], [0], [-7], [0], [-4], [11], [-25], [0], [0], [0], [26], [0], [2], [0], [11], [-16], [-28], [0], [-22], [0], [0], [17], [0], [0], [0], [0], [-1], [-31], [0], [-7], [0], [-16], [0], [-22], [0], [32], [5], [0], [-37], [0], [0], [-4], [-13], [8], [0], [0], [-34], [0], [38], [0], [-22], [0], [-37], [-13], [0], [0], [-10], [0], [-43], [0], [0], [-19], [0], [-43], [0], [0], [0], [-43], [29], [-40], [0], [0], [0], [-16], [-46], [0], [0], [0], [23], [29], [-10], [0], [17], [-1], [0], [47], [0], [0], [0], [38], [50], [0], [0], [-49], [0], [-22], [0], [-49], [-7], [53], [0], [-52], [-55], [0], [-49], [0], [56], [0], [0], [-37], [0], [47], [0], [53], [0], [23], [0], [-43], [0], [59], [0], [47], [0], [59], [0], [-52], [0], [26], [0], [0], [0], [41], [0], [0], [0], [-61], [59], [62], [0], [0], [47], [0], [5], [-52], [0], [41], [0], [65], [-49], [-55], [0], [-43], [0], [0], [0], [-19], [35], [29], [0], [62], [0], [53], [0], [0], [0], [2], [17], [0], [0], [0], [-19], [-70], [14], [0], [0], [71], [0], [0], [11], [71], [0], [-55], [0], [0], [-49], [-61], [0], [0], [0], [74], [68], [0], [53], [0], [-13], [0], [0], [41], [0], [2], [56], [29], [0], [-40], [0], [77], [0], [0], [0], [0], [71], [77], [11], [0], [0], [44], [0], [47], [0], [-79], [0], [0], [-58], [0], [0], [53], [-31], [0], [71], [-73], [0], [74], [-82], [0], [-64], [0], [0], [83], [0], [-49], [-19], [77], [-52], [14], [-55], [0], [23], [-73], [0], [0], [68], [0], [86], [-85], [0], [65], [0], [-4], [0], [0], [0], [0], [0], [62], [0], [-1], [0], [0], [-40], [-13], [0], [89], [0], [-88], [-34], [0], [-79], [0], [-73], [0], [0], [0], [23], [0], [-91], [0], [0], [5], [0], [32], [-85], [0], [-37], [0], [29], [-88], [8], [0], [0], [41], [0], [-91], [0], [-73], [0], [-82], [0], [-79], [20], [-37], [0], [0], [89], [0], [0], [74], [95], [0], [0], [41], [89], [0], [95], [-97], [0], [0], [0], [0], [0], [-1], [0], [0], [0], [11], [-73], [0], [47], [26], [0], [8], [0], [0], [-49], [-46], [0], [0], [-73], [0], [5], [0], [0], [71], [0], [56], [0], [59], [101], [-97], [0], [41], [0], [0], [-55], [0], [38], [-28], [0], [-61], [0], [83], [0], [-76], [0], [0], [-1], [-94], [0], [95], [0], [98], [0], [0], [-31], [65], [0], [0], [-4], [0], [0], [0], [71], [0], [0], [107], [0], [0], [0], [-109], [0]]
