
# q-expansion of newform 8112.2.a.m, downloaded from the LMFDB on 24 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 = 8112
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], [3], [-2], [-6], [0], [-3], [-2], [6], [3], [4], [-7], [-3], [10], [-6], [3], [0], [-7], [10], [-6], [-13], [4], [6], [18], [14], [15], [-14], [6], [14], [-3], [4], [0], [9], [4], [-9], [10], [5], [4], [0], [-6], [-6], [-7], [12], [23], [6], [10], [16], [-8], [-18], [-22], [-6], [-6], [-1], [12], [-3], [6], [18], [16], [17], [9], [-14], [-21], [10], [30], [-10], [3], [4], [23], [30], [-10], [-15], [-6], [-2], [29], [-20], [-24], [39], [14], [-3], [-1], [-24], [29], [6], [-13], [-14], [36], [18], [11], [15], [-38], [18], [0], [-2], [18], [-32], [-6], [3], [33], [34], [29], [34], [3], [0], [6], [22], [11], [0], [9], [-24], [-37], [-32], [-31], [-15], [-8], [-20], [-3], [16], [-24], [42], [-24], [5], [-13], [18], [48], [-26], [-30], [5], [0], [-14], [-31], [16], [36], [-14], [-34], [-30], [14], [-30], [28], [30], [-51], [4], [18], [40], [48], [17], [12], [-19], [21], [-26], [-18], [41], [33], [-8], [48], [-44], [24], [16], [33], [47], [42], [-24], [54], [22], [-60], [-3], [36], [-38], [5], [-34], [15], [42], [-31], [-36], [-37], [-32], [45], [-62], [3], [64], [-10], [-62], [0], [29], [-42], [-18], [-45], [-34], [28], [11], [30], [35], [30], [22], [-18], [6], [18], [-61], [5], [45], [12], [51], [40], [29], [-61], [66], [27], [-62], [-36], [9], [40], [-13], [-6], [-2], [-66], [0], [-1], [-8], [-27], [12], [-6], [29], [-20], [-51], [-20], [-18], [41], [57], [36], [-20], [-60], [-67], [4], [4], [21], [16], [-36], [-34], [-9], [24], [30], [18], [-74], [-14], [65], [-63], [36], [-44], [-36], [-26], [-6], [-34], [30], [-24], [-61], [30], [78], [38], [46], [-54], [-25], [-44], [-60], [17], [41], [9], [34], [30], [57], [-50], [51], [17], [52], [-73], [-68], [-58], [34], [-36], [53], [-49], [6], [-6], [40], [84], [53], [22], [42], [-37], [3], [34], [9], [-78], [-18], [9], [0], [14], [-69], [-8], [75], [12], [22], [-25], [-69], [-74], [-12], [-62], [47], [-78], [-31], [-84], [65], [-6], [-78], [-15], [-2], [-42], [-85], [42], [-30], [-34], [45], [82], [74], [3], [-2], [66], [-61], [-26], [46], [6], [39], [-55], [-57], [34], [18], [46], [30], [-31], [-6], [23], [28], [-10], [-87], [15], [82], [18], [-54], [-55], [-80], [30], [18], [-86], [38], [87], [58], [-31], [-3], [-42], [36], [-3], [-6], [89], [21], [0], [-84], [64], [38], [66], [-8], [-61], [12], [34], [-42], [51], [22], [-7], [18], [6], [-46], [-87], [-1], [-78], [21], [-2], [18], [-20], [-6], [82], [86], [-8], [84], [35], [-45], [18], [-26], [48], [62], [-38], [6], [4], [87], [-46], [-54], [-80], [33], [27], [-86], [50], [57], [94], [0], [11], [27], [30], [100], [-13], [-57], [12], [-14], [78], [36], [51], [29], [66], [54], [-58], [15], [84], [-15], [-98], [-84]]
