
# q-expansion of newform 945.2.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 = 945
weight = 2
poly_data = [-3, -1, 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, -1], [0, 0], [-1, 0], [1, 0], [-3, 0], [-1, 1], [0, 1], [-1, -1], [-3, 1], [-3, -1], [-7, 2], [-1, 2], [0, 3], [5, -4], [-3, -2], [-6, 5], [3, -4], [5, -7], [2, 5], [-9, 3], [-1, -4], [-10, -3], [3, 4], [3, -6], [2, -5], [0, 4], [5, -9], [-15, 0], [-4, 1], [-9, -3], [2, -9], [6, 1], [0, -8], [-1, -2], [6, -5], [-1, 0], [2, 2], [14, 4], [-15, -2], [3, 6], [-3, -6], [-1, 8], [9, -1], [-10, 7], [9, 6], [2, 9], [-16, 2], [11, 4], [18, -5], [11, -4], [3, 3], [-15, 4], [-16, 7], [-18, 6], [12, 0], [-24, 5], [9, -8], [-4, 7], [-10, 3], [0, 9], [-4, 3], [12, -14], [-4, 16], [-21, 3], [-1, -6], [15, -14], [11, 1], [11, -1], [-24, 2], [-1, 12], [-3, -9], [-15, 6], [-4, -1], [5, -3], [2, -12], [9, -4], [-12, 9], [2, 0], [-15, 9], [14, 0], [-27, 6], [5, -1], [-6, -13], [-10, 11], [-31, 2], [21, -9], [18, 0], [20, 1], [15, 5], [-19, 13], [0, -4], [-18, 13], [38, 0], [0, 9], [-13, -12], [-3, -7], [-9, 14], [39, 2], [-7, 9], [-4, -5], [29, 4], [27, -5], [-6, -13], [3, 12], [-10, 13], [35, 2], [9, -3], [21, 6], [-18, -8], [29, 7], [-13, 12], [5, -14], [-36, -3], [-7, 10], [-40, 4], [15, -9], [-25, 5], [-30, 0], [-21, 7], [36, -6], [-13, 9], [-28, 0], [27, -4], [-3, 21], [11, -8], [-3, -10], [-22, -10], [6, -11], [-31, 5], [-22, 15], [-34, 6], [0, -15], [14, 0], [20, -15], [21, -15], [-13, -6], [33, -15], [20, -10], [36, 3], [33, -6], [-13, -5], [3, -6], [5, 12], [27, -12], [23, -20], [18, -3], [-7, 16], [-21, 24], [14, 6], [-24, 24], [5, -12], [18, 12], [17, -10], [18, 2], [5, -25], [0, 12], [14, -5], [18, -25], [26, 8], [-21, 24], [24, -11], [33, 8], [11, -16], [24, -8], [18, -8], [-9, -5], [-13, -20], [-1, -6]]
