
# q-expansion of newform 3960.2.a.f, downloaded from the LMFDB on 23 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 = 3960
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], [-1], [1], [1], [-6], [-3], [-5], [2], [5], [5], [-1], [2], [12], [2], [13], [-2], [1], [16], [-15], [10], [2], [14], [-9], [-16], [10], [-16], [4], [10], [16], [-8], [-7], [-12], [-4], [15], [18], [7], [-1], [-1], [6], [-4], [22], [-12], [7], [24], [-1], [-1], [26], [-18], [6], [-11], [2], [18], [10], [22], [13], [-20], [24], [16], [-18], [10], [-6], [-22], [-21], [-30], [3], [-20], [29], [2], [10], [18], [-24], [-12], [14], [-2], [14], [2], [22], [21], [-32], [4], [-36], [8], [4], [16], [-12], [6], [33], [-5], [6], [-27], [-20], [-12], [13], [-8], [0], [-8], [26], [-4], [15], [-36], [30], [18], [-36], [-19], [-14], [39], [22], [27], [14], [-21], [-10], [-12], [0], [15], [9], [7], [6], [19], [-49], [-22], [-1], [0], [19], [34], [31], [46], [-1], [-26], [-32], [-8], [-33], [-7], [-2], [30], [10], [9], [-48], [-34], [6], [1], [-50], [44], [52], [4], [0], [22], [21], [-22], [46], [2], [-50], [5], [40], [-29], [-3], [-32], [-59], [10], [41], [-41], [-29], [55], [-44], [8], [14], [16], [8], [-6], [9], [21], [-22], [-10], [55], [-24], [53], [-4], [-16], [43], [5], [10], [33], [-15], [-47], [-38], [-30], [20], [-44], [-18], [36], [-56], [-20], [-46], [52], [-24], [-34], [24], [2], [-30], [-35], [5], [2], [-7], [-4], [-24], [33], [40], [0], [-18], [4], [-47], [60], [64], [-17], [60], [27], [-37], [58], [24], [47], [50], [14], [3], [-4], [56], [-63], [-66], [-27], [0], [-36], [-6], [-31], [-70], [10], [44], [-43], [19], [-30], [-48], [23], [-57], [12], [32], [-26], [2], [62], [-48], [30], [15], [-11], [-34], [0], [42], [19], [4], [-39], [72], [54], [18], [37], [24], [28], [-35], [-46], [60], [20], [-20], [-25], [44], [-18], [-6], [29], [-4], [14], [-77], [-26], [-24], [-42], [14], [27], [47], [4], [24], [-55], [68], [3], [14], [58], [-29], [10], [-48], [12], [41], [63], [30], [-22], [-16], [87], [20], [34], [65], [32], [40], [-36], [-22], [40], [28], [16], [79], [-8], [20], [-30], [-36], [15], [-33], [10], [10], [-12], [-22], [44], [-7], [-58], [-70], [50], [-45], [19], [30], [44], [-69], [-72], [79], [-42], [-44], [-38], [-69], [-3], [58], [93], [-19], [-43], [-78], [42], [-7], [-65], [29], [-45], [80], [17], [-60], [-61], [23], [2], [-20], [-45], [-31], [75], [15], [39], [54], [-60], [82], [60], [12], [-81], [-17], [24], [98], [29], [44], [-24], [80], [-56], [-9], [42], [-10], [60], [-78], [-13], [20], [-31], [-36], [-62], [-44], [59], [16], [0], [96], [49], [-24], [-18], [29], [-29], [24], [64], [-8], [86], [-62], [9], [-62], [55], [95], [-86], [-68], [-12], [6], [-6], [29], [2], [21], [-10], [-42], [48], [12], [20], [-78], [12], [12], [16], [58], [51], [68], [-4], [-76], [-91], [-38], [-29], [31], [20], [40], [2]]
