
# q-expansion of newform 8015.2.a.f, 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 = 8015
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 = [[2], [1], [1], [1], [-3], [3], [-1], [-6], [-2], [-3], [-2], [-4], [-6], [-8], [3], [2], [-8], [-10], [8], [12], [-2], [9], [-4], [12], [7], [8], [9], [14], [-1], [8], [-6], [-18], [-8], [-2], [-6], [15], [2], [4], [-15], [-9], [-12], [-6], [-13], [4], [10], [-20], [-23], [-13], [7], [-1], [-20], [25], [26], [30], [-30], [-18], [24], [22], [-2], [-23], [-3], [-9], [31], [20], [-17], [2], [24], [-28], [30], [2], [-15], [32], [-27], [-24], [12], [-24], [-33], [11], [-39], [-24], [0], [29], [5], [-38], [-14], [-6], [-1], [26], [14], [-10], [-1], [16], [-4], [19], [-13], [-33], [-26], [30], [8], [-29], [8], [-34], [-12], [6], [-20], [11], [36], [33], [-33], [8], [-1], [38], [-24], [44], [-7], [30], [-11], [32], [30], [-1], [32], [-12], [-5], [36], [-26], [-17], [13], [6], [-40], [-15], [-29], [16], [7], [-42], [22], [14], [-37], [37], [13], [27], [6], [-3], [4], [10], [-52], [-14], [-50], [26], [-8], [-48], [22], [2], [14], [8], [-24], [-36], [-9], [16], [-35], [28], [48], [14], [-34], [-14], [-12], [-39], [0], [-33], [-29], [-30], [44], [-8], [35], [-24], [46], [-24], [1], [-45], [-8], [40], [2], [14], [0], [50], [-24], [50], [-16], [7], [45], [-10], [10], [14], [15], [36], [-36], [37], [-61], [24], [57], [9], [-6], [-62], [-2], [38], [-42], [47], [-46], [-60], [6], [42], [32], [18], [-32], [-13], [-30], [0], [-48], [-6], [28], [-72], [14], [10], [-41], [-64], [61], [-26], [49], [11], [-13], [4], [38], [50], [0], [11], [-17], [27], [-72], [38], [-45], [-40], [-24], [54], [-31], [66], [-73], [-64], [8], [-38], [-59], [4], [58], [-12], [-12], [64], [-6], [5], [22], [60], [-65], [39], [16], [-34], [-4], [43], [-2], [-24], [15], [76], [-14], [6], [-36], [-58], [-75], [64], [78], [-19], [48], [6], [-37], [46], [-15], [67], [16], [28], [-25], [-13], [4], [42], [40], [-26], [-49], [-33], [24], [12], [-44], [58], [-28], [-17], [-70], [27], [-9], [0], [21], [-36], [29], [-36], [60], [50], [-72], [-32], [-68], [-59], [-30], [36], [-66], [88], [-6], [-25], [-19], [33], [72], [-28], [-66], [-36], [58], [-52], [47], [52], [54], [-86], [-14], [94], [40], [-7], [-4], [-17], [-61], [-62], [-68], [-27], [4], [68], [72], [-21], [6], [36], [-4], [-90], [-6], [33], [-96], [66], [-19], [31], [-55], [-58], [34], [12], [48], [52], [-80], [-56], [92], [73], [-56], [-58], [63], [34], [-53], [-63], [-2], [44], [5], [86], [-32], [72], [-73], [7], [-54], [-1], [-32], [-78], [26], [58], [-100], [21], [-25], [-11], [16], [3], [19], [-9], [31], [-52], [85], [30], [52], [-44], [81], [-74], [2], [66], [-16], [29], [0], [2], [-20], [-85], [56], [22], [-97], [2], [-42], [96], [-52], [-44], [-17], [16], [-66], [87], [81], [18], [78], [18], [75], [39], [74], [-18], [64], [-64]]
