
# q-expansion of newform 1170.2.a.m, downloaded from the LMFDB on 19 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 = 1170
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 = [[1], [0], [1], [0], [0], [-1], [6], [0], [4], [10], [0], [-6], [-2], [-4], [0], [6], [0], [6], [4], [-16], [-2], [0], [-4], [6], [14], [-6], [12], [4], [-14], [-10], [-12], [-12], [-6], [4], [14], [-16], [-14], [-4], [-24], [14], [20], [-10], [-24], [14], [22], [-16], [-12], [16], [-20], [2], [-18], [-16], [-14], [4], [6], [28], [-14], [-24], [2], [6], [-12], [-26], [-4], [16], [26], [-2], [8], [18], [-12], [34], [26], [-24], [20], [10], [24], [8], [18], [-14], [-18], [2], [-12], [-22], [24], [2], [-24], [4], [14], [6], [14], [-24], [28], [0], [8], [-20], [-40], [36], [38], [-10], [-20], [-22], [28], [30], [28], [6], [-4], [22], [12], [-6], [-24], [42], [-36], [-22], [-30], [40], [-48], [-2], [4], [36], [6], [20], [-30], [42], [30], [-12], [32], [18], [2], [-24], [12], [50], [-40], [48], [24], [2], [22], [34], [-26], [-36], [-18], [-26], [32], [22], [-44], [12], [54], [0], [-38], [38], [-20], [-48], [-14], [-34], [52], [36], [28], [-40], [-16], [46], [-22], [-10], [36], [-10], [-32], [-12], [2], [-16], [-8], [-22], [2], [-42], [-56], [10], [-48], [-58], [-16], [54], [0], [-2], [-52], [22], [-32], [52], [-46], [2], [8], [-38], [-14], [4], [-46], [48], [-22], [52], [36], [6], [52], [-34], [-38], [-46], [-54], [-44], [46], [40], [-22], [-62], [-32], [38], [16], [60], [22], [68], [50], [10], [-44], [-4], [-24], [-70], [44], [-18], [-64], [-50], [14], [40], [-26], [-64], [-28], [-50], [6], [-40], [28], [-32], [-62], [52], [24], [70], [52], [0], [58], [-42], [60], [24], [-60], [36], [-20], [10], [66], [0], [32], [-16], [40], [4], [2], [6], [-48], [58], [46], [24], [-2], [20], [6], [-34], [44], [4], [34], [-6], [-70], [-12], [30], [30], [44], [-18], [6], [68], [-26], [-72], [42], [32], [-76], [10], [34], [60], [-60], [40], [36], [-46], [84], [48], [-54], [6], [8], [-18], [50], [60], [-6], [40], [74], [-54], [0], [-2], [36], [-12], [-86], [-18], [88], [36], [20], [22], [-12], [-10], [8], [10], [28], [-18], [78], [76], [88], [-70], [-56], [56], [-10], [30], [-60], [-18], [-30], [72], [26], [50], [16], [20], [-52], [6], [18], [-10], [-56], [-28], [-48], [76], [66], [-6], [-14], [68], [-54], [-74], [6], [-32], [-66], [-36], [6], [36], [-40], [14], [-80], [-34], [-34], [44], [54], [46], [8], [-32], [-66], [48], [-6], [-42], [-84], [-24], [12], [-30], [14], [-56], [-38], [-4], [20], [-24], [34], [-32], [-54], [32], [48], [-26], [-18], [-46], [-90], [42], [24], [66], [80], [48], [-8], [-46], [52], [52], [-46], [-42], [-16], [-44], [-8], [66], [-8], [-26], [84], [-90], [66], [6], [-80], [18], [-18], [0], [34], [78], [44], [-8], [58], [22], [28], [-28], [-46], [-6], [-104], [-52], [-6], [12], [-30], [58], [-48], [36], [-18], [-18], [76], [-2], [-80], [-56]]
