
# q-expansion of newform 1980.2.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 = 1980
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], [2], [-1], [2], [0], [2], [0], [0], [8], [2], [0], [2], [0], [-6], [12], [2], [-4], [0], [2], [-10], [12], [6], [14], [0], [-4], [12], [-10], [-6], [2], [12], [-6], [-22], [12], [14], [14], [8], [24], [-12], [12], [-10], [0], [-10], [12], [-16], [-22], [20], [12], [2], [-12], [-12], [-10], [-12], [-18], [0], [-30], [2], [2], [0], [26], [-24], [-10], [-24], [-10], [-18], [8], [26], [12], [26], [-18], [0], [20], [-34], [-16], [-24], [-18], [2], [-6], [-10], [12], [-10], [-12], [-34], [2], [12], [-18], [26], [36], [-4], [-12], [-36], [8], [0], [-28], [24], [18], [-30], [-22], [-34], [26], [0], [-24], [-36], [-10], [-22], [36], [36], [-24], [26], [-22], [-34], [-6], [32], [-16], [30], [-40], [-24], [-6], [-12], [14], [-10], [36], [12], [8], [12], [2], [-24], [8], [50], [-34], [36], [32], [-34], [-12], [26], [18], [2], [6], [-36], [2], [24], [-4], [12], [2], [24], [-10], [-48], [-40], [24], [38], [18], [44], [36], [8], [24], [50], [-54], [14], [24], [-12], [-24], [-22], [-36], [18], [0], [8], [62], [-10], [-54], [60], [2], [48], [26], [20], [-30], [-10], [18], [26], [26], [-16], [-36], [14], [0], [-24], [18], [26], [56], [-10], [-12], [-46], [36], [-28], [-18], [-48], [-18], [-22], [62], [48], [36], [36], [-10], [-58], [62], [12], [-18], [20], [-48], [-12], [-52], [14], [42], [-40], [12], [-24], [62], [-58], [0], [-24], [42], [38], [-58], [66], [20], [72], [50], [-6], [24], [-10], [-48], [50], [-70], [-58], [-12], [44], [-12], [-22], [48], [36], [0], [-12], [14], [8], [-10], [-60], [-24], [-40], [-12], [38], [-12], [50], [-48], [-72], [-10], [-48], [-24], [26], [-10], [-18], [38], [-34], [-48], [14], [-10], [-42], [-28], [-6], [-30], [14], [36], [50], [-16], [26], [26], [-22], [56], [-12], [-22], [2], [-60], [24], [-40], [-60], [74], [-22], [-48], [-34], [-12], [8], [-24], [30], [-84], [12], [84], [-10], [-60], [-40], [-6], [0], [-10], [14], [0], [38], [-84], [68], [-10], [-60], [62], [0], [38], [-24], [78], [0], [44], [48], [62], [12], [24], [2], [12], [26], [-46], [-60], [-16], [-60], [2], [20], [68], [48], [12], [-10], [54], [14], [-12], [38], [-12], [2], [-60], [74], [-70], [-70], [-18], [84], [-52], [-30], [0], [-58], [32], [0], [-78], [-94], [-70], [90], [26], [50], [72], [-72], [-48], [48], [48], [-82], [0], [0], [0], [20], [-46], [48], [2], [2], [12], [80], [84], [0], [-22], [-10], [-12], [60], [-58], [-36], [2], [54], [-42], [74], [-24], [14], [48], [8], [-22], [74], [-72], [62], [18], [-36], [-52], [48], [-70], [14], [102], [56], [72], [50], [90], [2], [78], [36], [38], [-46], [96], [44], [84], [14], [48], [84], [-10], [50], [-90], [-84], [44], [78], [96], [-30], [-10], [24], [24], [74], [-102], [-12], [24], [-40], [-12]]
