
# q-expansion of newform 1110.2.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 = 1110
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], [-5], [-5], [-1], [-5], [-3], [3], [6], [-6], [-1], [0], [4], [0], [-3], [-10], [10], [-14], [6], [-9], [0], [5], [13], [-4], [-6], [-16], [15], [-5], [-10], [-3], [-20], [-2], [2], [2], [15], [-20], [-23], [5], [11], [8], [-10], [-7], [-2], [23], [-24], [-2], [-8], [8], [6], [-26], [12], [-18], [30], [15], [12], [9], [20], [25], [1], [5], [-11], [-30], [8], [24], [-10], [4], [-9], [-36], [24], [-26], [36], [-5], [14], [-26], [9], [34], [-36], [-27], [24], [-19], [-6], [-33], [21], [36], [-28], [30], [4], [26], [-28], [-12], [-21], [26], [-9], [-5], [-20], [15], [12], [-36], [5], [-43], [0], [22], [-13], [-10], [26], [-24], [-24], [44], [-13], [-18], [34], [-34], [-32], [-4], [0], [23], [3], [-48], [-20], [-47], [19], [-9], [48], [-22], [0], [-35], [-26], [-30], [46], [26], [10], [28], [37], [-18], [-14], [-21], [42], [-14], [-15], [40], [39], [-35], [12], [7], [-42], [19], [9], [7], [-12], [-52], [-40], [21], [-38], [37], [-8], [16], [2], [-34], [50], [4], [14], [-18], [-4], [39], [-58], [8], [-39], [14], [36], [42], [14], [32], [7], [19], [-10], [32], [23], [-16], [26], [44], [42], [43], [30], [24], [57], [36], [59], [12], [-52], [-6], [60], [25], [-47], [12], [-32], [40], [14], [-48], [29], [-22], [-37], [40], [10], [20], [-34], [7], [-29], [-19], [-24], [10], [16], [-4], [63], [-31], [-27], [58], [10], [-57], [-9], [-54], [-49], [-18], [46], [4], [-31], [0], [20], [-28], [-12], [-14], [-56], [-35], [-8], [36], [-30], [-59], [-38], [6], [-24], [36], [-11], [-35], [-35], [0], [-15], [50], [-38], [-26], [9], [-13], [4], [-54], [56], [42], [32], [-62], [26], [-79], [38], [-70], [0], [40], [-34], [63], [44], [61], [-42], [18], [26], [25], [1], [64], [-48], [-59], [-71], [57], [46], [-77], [-6], [38], [-22], [-42], [0], [-44], [66], [-42], [10], [66], [-75], [-18], [50], [26], [37], [44], [-78], [-65], [-27], [-64], [36], [-42], [-18], [24], [-66], [11], [32], [24], [46], [50], [39], [60], [79], [-6], [-68], [13], [-66], [53], [-37], [-46], [75], [-37], [-37], [-8], [7], [20], [72], [16], [56], [33], [-42], [8], [16], [50], [6], [-65], [16], [58], [-22], [-10], [-24], [-60], [38], [-36], [-12], [36], [66], [16], [82], [-16], [-26], [-90], [-91], [-12], [-42], [46], [4], [-19], [-14], [-96], [80], [38], [-22], [-51], [-56], [-62], [-14], [-32], [86], [-82], [-12], [88], [-22], [-63], [-14], [-58], [-93], [78], [-21], [47], [71], [-78], [71], [-38], [-2], [-4], [24], [-73], [42], [84], [-39], [13], [54], [52], [-7], [-24], [-13], [-10], [12], [82], [-101], [-32], [-49], [24], [34], [18], [7], [-43], [-48], [-56], [76], [-55], [-19], [-18], [79], [80], [-65], [32], [37], [14], [48], [-98], [3], [-6], [15], [-29], [42], [-9], [50], [42], [76]]
