
# q-expansion of newform 5610.2.a.f, downloaded from the LMFDB on 20 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 = 5610
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], [-3], [-1], [-4], [-1], [4], [-1], [-3], [-5], [-6], [8], [13], [-12], [-4], [-6], [-10], [8], [14], [-6], [4], [-8], [-4], [-1], [-2], [-11], [5], [8], [10], [-6], [12], [17], [1], [16], [12], [-18], [21], [18], [-6], [-6], [-5], [-9], [-6], [-12], [8], [15], [-19], [21], [-24], [23], [6], [-9], [0], [-17], [11], [26], [-25], [-2], [3], [-2], [-3], [20], [0], [17], [-31], [-11], [18], [12], [26], [25], [14], [22], [22], [-8], [8], [22], [-18], [3], [8], [25], [8], [35], [12], [-34], [13], [41], [-22], [-26], [-24], [12], [29], [2], [8], [4], [28], [30], [-18], [-11], [-28], [2], [-5], [-12], [34], [44], [38], [27], [-6], [1], [-46], [40], [10], [-6], [14], [-22], [-27], [11], [22], [-39], [-23], [-16], [-8], [18], [12], [12], [46], [50], [-6], [5], [44], [-10], [8], [45], [-23], [-3], [2], [48], [-12], [-52], [-54], [8], [-33], [10], [-37], [34], [18], [-51], [-7], [35], [32], [13], [19], [-24], [24], [-37], [-32], [1], [-7], [-38], [46], [18], [-38], [-42], [-12], [2], [-41], [-47], [7], [-15], [-39], [47], [12], [-4], [27], [36], [50], [-28], [-10], [52], [-53], [21], [-12], [-46], [-10], [0], [26], [61], [-24], [-13], [-8], [19], [-28], [-65], [-45], [-31], [4], [10], [-34], [-33], [-19], [46], [14], [-11], [-40], [30], [60], [-34], [-24], [-48], [-23], [52], [-27], [4], [-23], [24], [-8], [-6], [-43], [48], [30], [-13], [44], [-28], [-66], [0], [-11], [6], [-38], [50], [-10], [49], [23], [-21], [-13], [-24], [25], [-8], [64], [15], [-16], [-64], [-66], [-55], [62], [65], [29], [-58], [19], [-14], [-61], [64], [0], [-60], [-50], [-17], [-49], [10], [42], [21], [42], [60], [10], [46], [19], [-28], [-35], [43], [0], [78], [23], [-7], [2], [-74], [-49], [-21], [52], [-12], [13], [38], [-60], [51], [8], [-48], [37], [-38], [32], [14], [-10], [-34], [63], [62], [-16], [-36], [-54], [25], [-18], [-18], [0], [-18], [-77], [31], [-11], [26], [84], [-60], [62], [77], [14], [59], [-6], [-74], [-2], [-20], [60], [-13], [-56], [-80], [-40], [22], [5], [-35], [-23], [-36], [31], [-58], [-41], [80], [61], [36], [-36], [30], [25], [6], [24], [12], [68], [-56], [37], [12], [34], [-10], [-54], [39], [62], [-76], [-6], [-10], [-52], [-27], [-35], [8], [-6], [-13], [12], [-6], [18], [-21], [84], [-29], [-2], [-63], [18], [-76], [62], [4], [-16], [-30], [-10], [-70], [2], [-4], [6], [94], [-35], [-28], [4], [45], [-65], [-14], [-38], [56], [39], [-8], [27], [-1], [91], [70], [53], [50], [96], [-28], [-9], [94], [102], [64], [-24], [4], [30], [50], [-36], [-76], [74], [48], [-72], [30], [-2], [82], [-58], [-26], [-75], [-26], [66], [12], [1], [63], [-46], [24], [8], [-36], [-40], [102], [-54], [48], [68], [26], [-42], [48], [-70], [77], [-13]]
