
# q-expansion of newform 1155.2.a.f, downloaded from the LMFDB on 23 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 = 1155
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], [1], [1], [1], [-1], [-2], [2], [4], [0], [6], [0], [6], [-6], [-4], [0], [-2], [4], [6], [12], [0], [10], [8], [-4], [10], [10], [-10], [8], [4], [-2], [-6], [16], [-12], [2], [-4], [-18], [-16], [22], [-4], [-16], [14], [-4], [-10], [24], [2], [-10], [8], [-20], [-16], [12], [22], [-6], [-24], [10], [-12], [-14], [0], [-18], [-16], [-10], [-30], [-20], [6], [-12], [-32], [-14], [22], [-4], [18], [-12], [-26], [2], [16], [-16], [22], [-4], [-16], [6], [22], [34], [-14], [12], [6], [8], [-6], [24], [36], [2], [-22], [-2], [24], [-12], [-32], [16], [20], [-28], [0], [-2], [-6], [-20], [-2], [4], [30], [-36], [2], [-12], [10], [-36], [2], [0], [2], [-48], [6], [18], [12], [24], [18], [20], [-24], [38], [-20], [-10], [34], [6], [4], [36], [22], [-26], [24], [24], [-34], [12], [-16], [-32], [38], [10], [26], [-42], [20], [-18], [18], [-36], [14], [16], [-28], [14], [0], [22], [10], [-20], [24], [-18], [-30], [-36], [-48], [36], [-40], [16], [-14], [26], [-50], [-36], [42], [-40], [-12], [-38], [-56], [-16], [6], [34], [6], [52], [14], [-48], [10], [-16], [42], [-44], [54], [24], [54], [-24], [52], [-10], [-6], [-24], [22], [62], [-44], [-54], [-48], [42], [-12], [52], [-2], [12], [-54], [-30], [46], [18], [0], [54], [0], [-34], [42], [-12], [-18], [0], [-4], [-46], [-52], [-14], [-10], [-16], [12], [72], [-38], [64], [-46], [48], [54], [22], [-24], [-30], [-8], [-68], [-10], [42], [-8], [8], [20], [-34], [-44], [-24], [-14], [44], [8], [34], [22], [-12], [-16], [-4], [-20], [40], [-50], [34], [8], [0], [-36], [68], [-24], [30], [-30], [16], [-54], [14], [-36], [54], [-68], [22], [-22], [64], [-36], [-2], [46], [66], [-60], [78], [42], [76], [-58], [-2], [28], [-62], [56], [66], [-40], [52], [46], [-22], [-28], [8], [24], [-80], [46], [-52], [-56], [-62], [54], [56], [-46], [-2], [-44], [-86], [12], [-18], [-18], [16], [54], [12], [52], [26], [62], [-40], [-84], [-4], [34], [-28], [-82], [0], [38], [-40], [-58], [-6], [-52], [16], [-46], [60], [8], [10], [-38], [-44], [10], [78], [88], [-6], [-62], [-44], [44], [72], [22], [78], [38], [16], [-4], [4], [-52], [-2], [-78], [58], [-48], [-90], [-14], [38], [-24], [-10], [-36], [-10], [-28], [48], [70], [68], [26], [-18], [80], [6], [-22], [-8], [-52], [-62], [-48], [6], [-6], [8], [-28], [84], [-38], [46], [56], [58], [28], [-20], [-40], [30], [-24], [-2], [76], [-88], [-14], [50], [-86], [62], [66], [24], [2], [-28], [24], [-48], [-98], [12], [48], [-62], [-26], [52], [4], [48], [-38], [-32], [-102], [92], [94], [-42], [74], [80], [-22], [-58], [-56], [-82], [26], [52], [92], [-62], [22], [-76], [28], [42], [78], [-40], [-40], [-14], [-64], [-82], [-10], [8], [60], [-62], [62], [76], [82], [12], [8]]
