
# q-expansion of newform 7056.2.a.f, downloaded from the LMFDB on 17 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 = 7056
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], [0], [-3], [0], [-3], [2], [-3], [1], [3], [6], [7], [-1], [-6], [4], [-9], [-3], [9], [-1], [7], [0], [-1], [13], [12], [-15], [-10], [-15], [-11], [15], [-1], [-6], [-8], [3], [21], [-20], [-3], [-17], [-13], [-11], [-12], [9], [21], [-10], [-9], [11], [-18], [7], [4], [-8], [-3], [11], [21], [-12], [-1], [0], [-3], [-3], [-3], [-11], [-13], [-30], [-29], [-6], [28], [-27], [23], [9], [13], [-34], [9], [26], [21], [15], [-5], [-25], [-8], [-33], [-15], [-37], [-3], [11], [-12], [-22], [-15], [-10], [1], [-9], [18], [23], [6], [16], [-21], [-3], [19], [24], [-11], [0], [-3], [-39], [1], [35], [-8], [33], [9], [9], [-29], [-1], [12], [21], [-27], [14], [-47], [-25], [-6], [31], [16], [-15], [-20], [21], [-39], [12], [11], [14], [-27], [21], [13], [18], [-1], [-21], [-32], [-25], [19], [-48], [25], [2], [-3], [-34], [33], [31], [-42], [33], [52], [-27], [-5], [-36], [-1], [-24], [-22], [-15], [-23], [51], [-13], [-18], [-44], [39], [-41], [48], [1], [9], [26], [-27], [-45], [18], [16], [51], [-27], [33], [19], [59], [-10], [33], [-39], [50], [3], [23], [7], [-6], [-32], [-3], [16], [-13], [13], [12], [38], [-15], [-27], [21], [47], [-5], [11], [-15], [-1], [-60], [-47], [-27], [21], [45], [11], [-37], [-30], [-21], [9], [40], [23], [35], [36], [-3], [61], [39], [6], [-41], [-37], [30], [-8], [27], [-63], [-13], [31], [-51], [-57], [66], [-61], [-20], [-39], [25], [60], [26], [45], [21], [-23], [-45], [-61], [-65], [16], [-27], [4], [-3], [11], [21], [60], [-36], [-39], [1], [-29], [47], [30], [-45], [-8], [45], [-5], [-36], [2], [-15], [9], [14], [21], [-57], [-37], [67], [54], [-61], [-5], [-60], [-25], [26], [45], [-23], [-18], [42], [4], [-63], [-61], [-29], [59], [-47], [-58], [-11], [3], [-13], [71], [63], [-63], [-65], [-24], [50], [49], [15], [-1], [30], [67], [-54], [57], [21], [33], [24], [50], [-63], [-23], [6], [51], [40], [23], [-51], [-65], [60], [25], [62], [33], [-70], [39], [35], [-45], [-75], [45], [31], [72], [35], [12], [-63], [14], [-6], [67], [11], [-90], [28], [-87], [71], [37], [13], [27], [54], [23], [9], [40], [-39], [19], [-36], [2], [-3], [-70], [37], [23], [-30], [-42], [-32], [-63], [-36], [-37], [-71], [12], [57], [85], [11], [-42], [19], [-25], [-90], [3], [21], [-3], [-24], [-46], [-27], [-87], [27], [55], [35], [78], [-89], [-70], [33], [-71], [39], [42], [-65], [-73], [-27], [0], [11], [-87], [14], [-87], [-6], [-56], [57], [-68], [57], [-5], [-85], [37], [96], [50], [-27], [-87], [85], [-9], [-49], [-29], [42], [52], [9], [23], [-51], [-95], [33], [-39], [-35], [-37], [-30], [-29], [99], [71], [9], [-36], [73], [38], [-15], [15], [-41], [18], [-105], [-51], [-1], [48], [-12], [38], [-27], [-81], [18], [-65], [33]]
