
# q-expansion of newform 1350.2.a.f, 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 = 1350
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], [0], [-1], [0], [2], [-6], [-1], [-6], [6], [5], [-7], [-12], [11], [-12], [0], [6], [-7], [-4], [-6], [-7], [-1], [6], [6], [5], [12], [5], [-6], [-13], [-12], [-16], [-18], [-12], [-13], [-18], [-7], [17], [-4], [18], [6], [0], [-22], [24], [-1], [12], [8], [20], [17], [12], [-13], [6], [-24], [2], [30], [24], [18], [-6], [11], [-1], [12], [5], [6], [-19], [-12], [-34], [6], [35], [26], [6], [-19], [24], [6], [-28], [29], [32], [-6], [0], [-7], [24], [26], [-18], [-1], [-18], [-13], [5], [-30], [-30], [26], [-6], [-31], [-6], [18], [8], [24], [-7], [6], [24], [-12], [23], [-25], [35], [-42], [24], [-30], [23], [23], [-6], [-6], [48], [11], [5], [-43], [-36], [11], [-4], [-30], [-4], [0], [-24], [30], [-31], [-37], [-24], [-6], [-28], [6], [-49], [-30], [5], [14], [20], [24], [23], [23], [12], [38], [18], [-37], [-36], [42], [29], [-36], [-16], [-12], [-25], [-54], [26], [36], [-13], [-54], [-31], [-18], [-19], [-36], [17], [-6], [-37], [54], [-25], [36], [-18], [-24], [-13], [-24], [-6], [-24], [23], [-10], [41], [54], [-42], [-7], [48], [-19], [-16], [-42], [17], [6], [8], [-43], [-16], [36], [47], [18], [6], [-36], [-34], [56], [29], [18], [-19], [30], [-43], [0], [60], [30], [-49], [-1], [-12], [36], [12], [-37], [-13], [17], [-24], [66], [-55], [-6], [-6], [8], [50], [54], [-28], [48], [-54], [-1], [-19], [30], [-48], [-24], [-61], [-61], [42], [20], [12], [-22], [36], [18], [-16], [72], [59], [-16], [35], [-36], [35], [-12], [5], [-60], [-36], [54], [0], [-52], [-55], [-34], [-54], [54], [17], [-48], [-13], [42], [-22], [-6], [12], [-10], [36], [-60], [-13], [44], [-18], [38], [-49], [-36], [59], [65], [12], [-28], [18], [-18], [-4], [-54], [5], [11], [-46], [-4], [5], [23], [-54], [-55], [83], [30], [24], [-76], [-42], [26], [-76], [-30], [26], [-54], [53], [18], [42], [-54], [-12], [-66], [41], [72], [-13], [30], [78], [17], [23], [-60], [5], [-78], [23], [5], [-60], [-37], [-66], [29], [36], [18], [-6], [-79], [48], [77], [-18], [-6], [-55], [-6], [53], [-34], [48], [65], [36], [29], [-19], [41], [-6], [-42], [26], [-12], [56], [66], [-19], [-12], [5], [-60], [-19], [56], [-31], [-18], [-60], [-91], [-90], [-42], [35], [29], [36], [-48], [-85], [29], [-60], [-13], [-22], [-78], [-12], [-12], [30], [-48], [-67], [60], [-36], [0], [47], [-37], [-54], [-79], [-10], [-18], [65], [-72], [66], [-67], [35], [30], [18], [-10], [-36], [-7], [-18], [30], [-25], [48], [5], [-18], [-67], [23], [11], [-30], [53], [60], [-30], [92], [60], [-19], [-40], [-60], [35], [54], [-58], [48], [23], [-18], [6], [-16], [59], [48], [-76], [-6], [71], [90], [-24], [-64], [41], [-66], [-102], [-31], [54], [-36], [24], [-31], [12], [30], [-34], [42], [-48], [-66], [35], [-36]]
