
# q-expansion of newform 1200.2.a.f, downloaded from the LMFDB on 19 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 = 1200
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], [0], [1], [-6], [5], [-6], [-5], [6], [-6], [1], [2], [0], [1], [-6], [-12], [6], [-13], [-11], [0], [2], [-8], [6], [0], [-7], [-12], [4], [-12], [-7], [12], [16], [-12], [-18], [4], [18], [-5], [23], [-5], [-12], [6], [6], [5], [6], [11], [6], [7], [-5], [25], [0], [-19], [0], [12], [-7], [12], [12], [-24], [6], [16], [-7], [6], [-17], [-18], [-17], [6], [11], [24], [28], [17], [18], [26], [-18], [-24], [31], [-1], [-23], [-12], [-24], [-13], [36], [-19], [-12], [-10], [-18], [11], [-23], [24], [-12], [-10], [36], [-32], [18], [6], [7], [-24], [-29], [24], [-18], [-18], [-17], [-7], [-8], [18], [30], [6], [-5], [-37], [36], [24], [-36], [-37], [-32], [2], [36], [31], [13], [-36], [4], [-12], [-6], [-48], [-22], [-10], [-6], [24], [-8], [6], [29], [-18], [-23], [14], [-20], [-36], [-32], [-19], [12], [-13], [48], [7], [-12], [36], [-5], [6], [13], [12], [2], [-54], [23], [-12], [40], [-48], [-49], [0], [-23], [6], [28], [-18], [-53], [-42], [47], [30], [-42], [-24], [40], [42], [18], [-12], [37], [26], [-19], [-6], [42], [47], [-42], [53], [31], [48], [-29], [30], [25], [-1], [-11], [24], [-43], [-6], [-30], [30], [-19], [43], [53], [66], [-31], [24], [-11], [54], [18], [-66], [-58], [-10], [-6], [30], [-24], [-44], [-22], [26], [-36], [-6], [31], [66], [-36], [37], [-13], [-30], [-59], [36], [-6], [-10], [-23], [66], [-48], [54], [17], [-29], [-30], [-56], [0], [-1], [18], [18], [7], [-30], [-31], [-20], [-5], [-42], [28], [24], [50], [-36], [18], [-12], [-6], [37], [7], [-37], [6], [-18], [-5], [-6], [-35], [-42], [-58], [42], [12], [-67], [-78], [-12], [17], [-44], [-60], [-34], [61], [-18], [-58], [-25], [-6], [-8], [-78], [-72], [76], [18], [-49], [-56], [26], [-11], [-55], [37], [-66], [-25], [23], [48], [24], [-32], [-6], [-13], [79], [42], [-79], [24], [64], [24], [-54], [-12], [60], [24], [-13], [-24], [25], [0], [-54], [-44], [-58], [-12], [-83], [-54], [4], [-43], [-36], [65], [36], [38], [-6], [60], [60], [-35], [54], [-7], [-6], [12], [65], [6], [-80], [-13], [-36], [49], [18], [74], [91], [67], [-30], [-48], [-31], [54], [61], [-90], [-53], [36], [-37], [-36], [77], [88], [11], [0], [18], [-17], [6], [78], [17], [19], [-60], [12], [-5], [74], [6], [-77], [-25], [-42], [48], [42], [-30], [-18], [47], [54], [90], [48], [-20], [35], [-72], [-92], [-25], [60], [-11], [-6], [12], [-32], [5], [-24], [-78], [47], [-60], [5], [-66], [-66], [16], [48], [76], [42], [-23], [86], [52], [18], [29], [-24], [60], [-83], [90], [-7], [-95], [-36], [13], [12], [-22], [60], [-17], [-60], [84], [16], [-43], [48], [43], [-36], [47], [-84], [-6], [-5], [5], [60], [66], [7], [48], [12], [-24], [11], [54], [-102], [-34], [72], [-72], [-12], [-35], [-30]]
