
# q-expansion of newform 2240.2.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 = 2240
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 = [[0], [-1], [-1], [-1], [3], [7], [-5], [-4], [-4], [5], [-2], [0], [6], [-8], [9], [0], [-12], [6], [-16], [-4], [-14], [-9], [8], [-6], [3], [0], [-15], [-10], [-9], [-8], [-20], [0], [-14], [-6], [6], [15], [2], [10], [-15], [21], [24], [-24], [-11], [-14], [-2], [8], [13], [1], [-25], [-14], [-8], [-9], [28], [6], [6], [-10], [18], [14], [-30], [-9], [-15], [25], [13], [12], [1], [-18], [28], [-22], [12], [-16], [7], [-32], [21], [26], [-8], [12], [-5], [-7], [-3], [22], [16], [-13], [33], [-14], [2], [24], [15], [-2], [40], [-10], [17], [32], [-30], [-35], [-21], [-9], [6], [0], [4], [-11], [22], [32], [-4], [-26], [-20], [-11], [-12], [-19], [25], [-6], [-17], [-26], [-38], [-4], [1], [-42], [23], [28], [-24], [31], [-28], [26], [-21], [18], [-8], [13], [21], [28], [0], [-13], [-5], [4], [19], [52], [-48], [-6], [11], [-49], [-19], [-39], [10], [43], [14], [-26], [-14], [-4], [-42], [58], [-20], [2], [0], [8], [-58], [-8], [12], [48], [35], [44], [47], [46], [-18], [34], [-42], [30], [44], [-45], [-16], [41], [-47], [-2], [-20], [-2], [-39], [34], [-8], [-4], [31], [-13], [-64], [26], [54], [-62], [-16], [-42], [-38], [-16], [40], [-57], [53], [-26], [14], [4], [-51], [-18], [16], [57], [-37], [-22], [-5], [-63], [14], [6], [14], [16], [-56], [65], [-38], [-34], [10], [50], [-70], [-18], [34], [27], [50], [26], [44], [-30], [64], [14], [22], [30], [-55], [38], [19], [-38], [-9], [-51], [17], [12], [34], [-30], [8], [41], [-1], [-19], [34], [-4], [1], [28], [16], [-12], [25], [-38], [-39], [24], [-44], [62], [73], [4], [-44], [-70], [12], [44], [18], [47], [50], [28], [63], [-63], [-8], [-26], [-12], [-5], [26], [-16], [47], [6], [24], [-8], [46], [38], [-77], [-56], [-2], [9], [12], [-22], [-79], [-34], [-19], [71], [56], [14], [-29], [73], [52], [-16], [0], [-80], [-29], [-43], [34], [30], [44], [26], [-50], [-21], [56], [19], [19], [-50], [-47], [24], [57], [-72], [84], [-40], [72], [80], [8], [57], [-6], [-54], [0], [-50], [34], [33], [-17], [29], [-14], [30], [-70], [6], [6], [60], [-71], [60], [78], [-22], [-86], [-2], [-6], [-43], [20], [27], [15], [26], [16], [5], [-22], [52], [-80], [-55], [-28], [28], [-18], [22], [-36], [-67], [60], [10], [-45], [53], [47], [-10], [-50], [-60], [-12], [-66], [-78], [36], [-50], [39], [-20], [36], [-35], [36], [-55], [3], [-46], [36], [33], [-70], [38], [84], [-15], [49], [-26], [89], [-54], [-26], [88], [70], [86], [-19], [53], [-17], [-8], [63], [-3], [49], [33], [28], [-99], [-84], [-94], [-54], [-57], [70], [-22], [88], [-76], [-21], [12], [38], [-6], [-81], [20], [-68], [71], [-66], [-12], [28], [-16], [-48], [37], [8], [74], [-63], [-13], [22], [2], [-72], [-51], [25], [0], [-86], [62], [90]]
