
# q-expansion of newform 3969.2.a.f, downloaded from the LMFDB on 25 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 = 3969
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], [0], [1], [0], [5], [5], [-3], [-1], [3], [-1], [0], [3], [5], [-1], [0], [-9], [0], [14], [4], [-12], [-3], [8], [9], [13], [9], [17], [1], [17], [-9], [-1], [-12], [1], [-9], [-9], [3], [5], [14], [-11], [19], [14], [19], [14], [8], [-10], [2], [-3], [13], [-19], [3], [1], [3], [-15], [-11], [28], [29], [5], [-3], [-1], [19], [-29], [-28], [5], [-28], [0], [-14], [-6], [8], [-29], [4], [-19], [-11], [-11], [3], [-25], [-12], [-27], [-9], [-15], [3], [-14], [-9], [-1], [-9], [14], [0], [36], [30], [22], [-19], [13], [27], [-25], [19], [13], [31], [0], [29], [-3], [-1], [-25], [-29], [-37], [28], [-34], [32], [-31], [37], [-15], [-24], [9], [1], [19], [27], [-25], [-40], [-9], [19], [-31], [3], [27], [14], [-29], [-42], [-9], [28], [30], [-6], [27], [-47], [-27], [-9], [-15], [31], [2], [-27], [-23], [-31], [-28], [-23], [-9], [28], [22], [-24], [-12], [25], [37], [37], [-11], [1], [-39], [-53], [42], [44], [29], [5], [27], [17], [14], [-42], [14], [-20], [-26], [13], [-57], [18], [3], [-37], [17], [-26], [-55], [-51], [33], [3], [47], [-56], [5], [55], [-37], [0], [14], [8], [9], [-1], [-43], [-24], [17], [19], [59], [-53], [25], [41], [27], [17], [-42], [-37], [-42], [-10], [-6], [9], [-57], [-38], [37], [-27], [-67], [-28], [53], [-57], [3], [-57], [56], [3], [33], [-29], [28], [56], [-70], [-52], [-11], [17], [27], [59], [47], [31], [-67], [-37], [-1], [-14], [19], [-56], [36], [46], [-31], [41], [-10], [9], [3], [-55], [31], [-15], [56], [33], [-57], [0], [31], [33], [28], [-28], [-28], [4], [-12], [-1], [69], [32], [-19], [-11], [-11], [47], [3], [-42], [13], [5], [55], [70], [70], [-70], [28], [-54], [37], [13], [74], [-55], [47], [17], [-25], [5], [55], [-25], [-9], [-53], [56], [56], [60], [-19], [-19], [1], [-39], [19], [27], [31], [42], [47], [25], [3], [75], [-29], [70], [-28], [-19], [-1], [-47], [57], [3], [19], [41], [3], [30], [33], [33], [-25], [31], [0], [-65], [78], [60], [16], [-59], [9], [-24], [-19], [-57], [-53], [31], [-42], [-40], [75], [-43], [73], [28], [-20], [-54], [-62], [-66], [-47], [3], [-65], [-56], [2], [57], [-23], [-85], [-65], [-29], [-51], [-29], [-62], [44], [-31], [-20], [9], [41], [29], [-9], [-85], [87], [31], [-70], [57], [-3], [-53], [-57], [27], [70], [75], [-20], [-59], [22], [65], [-65], [30], [-51], [27], [-81], [27], [28], [-53], [28], [-40], [25], [-99], [93], [-67], [-57], [100], [-65], [-9], [53], [89], [14], [-67], [-9], [55], [42], [75], [28], [-20], [-38], [-31], [-75], [-15], [18], [85], [31], [31], [-27], [-87], [83], [-93], [-29], [28], [-84], [-98], [22], [100], [-20], [-57], [-15], [45], [-53], [42], [-27], [-9], [-43], [55], [47], [-14], [-42], [-76], [30], [-87], [87]]
