
# q-expansion of newform 950.2.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 = 950
weight = 2
poly_data = [-2, 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 = [[-1, 0], [1, 1], [0, 0], [3, -1], [0, 1], [3, -2], [1, 0], [-1, 0], [5, 3], [-3, -2], [2, -3], [0, 6], [0, 3], [6, 3], [0, 0], [-3, -6], [-3, -7], [10, -3], [9, 3], [-12, 1], [3, -6], [2, 6], [6, -6], [0, -5], [-6, -4], [6, -5], [0, -3], [7, -9], [-1, 12], [-6, 3], [6, 6], [0, 12], [13, 0], [-12, 0], [12, -4], [-2, 6], [6, -4], [-6, 3], [-14, 3], [-8, -6], [-6, 4], [0, 6], [-3, -11], [-6, -4], [-16, -3], [5, -3], [-7, 9], [-18, -2], [-21, -3], [-2, -12], [-8, 12], [3, 7], [-8, -12], [-12, -11], [8, -9], [-10, 12], [6, 16], [-23, 3], [0, 0], [0, -3], [18, -10], [3, 6], [-12, 4], [-9, -3], [9, 12], [-1, 6], [-15, 3], [24, 7], [-14, -6], [6, 0], [-11, -6], [-27, -3], [0, 18], [9, 0], [7, -3], [8, -3], [30, 5], [24, 0], [-24, 1], [4, -15], [18, -1], [11, -6], [12, 13], [6, 11], [-10, -3], [0, -3], [6, -8], [-3, 0], [12, 11], [0, -10], [-12, -9], [0, 22], [18, -14], [-18, 11], [-30, -3], [27, 9], [12, -12], [12, -8], [-15, -9], [14, 3], [-6, 8], [16, 0], [-12, -12], [0, 20], [2, 3], [-9, 8], [26, -3], [22, 0], [6, 15], [-16, 3], [6, 2], [18, 21], [4, 6], [12, 3], [-22, 12], [6, 17], [-6, 6], [-23, -3], [-12, 12], [9, -7], [7, 18], [-12, 0], [-7, 24], [12, 0], [14, -6], [0, -12], [-34, 0], [-15, -7], [-3, -9], [12, 0], [14, 9], [-6, -9], [-14, 9], [18, 4], [27, -12], [-11, 18], [-3, -12], [-3, 27], [-39, 6], [-9, 14], [23, -21], [12, 4], [-3, 9], [9, 21], [-15, -18], [0, -16], [6, 24], [0, 0], [-8, -9], [-12, -9], [-33, -10], [0, 28], [6, -6], [-24, 15], [9, -21], [12, -20], [15, 15], [-15, -2], [27, 0], [39, -10], [-12, 0], [6, -9], [12, -22], [36, -4], [-4, -3], [-26, 18], [18, 3], [36, -9]]
