
# q-expansion of newform 2310.2.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 = 2310
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], [1], [-1], [-1], [1], [-2], [-2], [4], [-4], [-6], [-4], [-2], [10], [4], [-8], [-6], [-4], [-2], [-12], [-8], [2], [4], [0], [-10], [-2], [-10], [-8], [-4], [6], [6], [8], [-12], [-18], [-4], [2], [-4], [2], [12], [-12], [-22], [20], [-2], [-16], [22], [-6], [4], [8], [8], [0], [6], [6], [0], [2], [20], [-18], [16], [-6], [-24], [2], [-18], [-28], [18], [-20], [16], [-10], [2], [4], [-18], [-20], [30], [-18], [32], [-8], [10], [-12], [-24], [14], [-38], [-30], [10], [12], [6], [16], [6], [8], [16], [34], [22], [22], [16], [20], [40], [40], [-4], [20], [12], [10], [6], [-20], [22], [36], [18], [0], [-10], [0], [22], [4], [6], [32], [26], [0], [34], [6], [24], [8], [-30], [-4], [24], [-30], [-36], [-18], [-2], [-6], [0], [-48], [-14], [-26], [40], [-8], [6], [40], [0], [8], [-18], [42], [-30], [26], [-20], [-14], [-34], [-20], [10], [-16], [12], [-42], [40], [-34], [-10], [-40], [44], [-38], [30], [20], [-28], [28], [48], [-28], [-42], [26], [-58], [0], [6], [-24], [-20], [-18], [-24], [0], [-2], [2], [42], [60], [-2], [-24], [46], [4], [-26], [24], [-58], [0], [-18], [-24], [36], [-18], [-18], [36], [10], [-30], [-36], [-14], [32], [14], [-36], [20], [-54], [-52], [-58], [10], [-18], [6], [36], [10], [-40], [-22], [-62], [20], [18], [-20], [68], [30], [48], [22], [-46], [-24], [4], [32], [-62], [32], [-46], [20], [18], [6], [-32], [18], [24], [-32], [6], [6], [56], [32], [4], [-58], [-36], [-68], [46], [-60], [-36], [-6], [-6], [-60], [40], [16], [12], [24], [-2], [-2], [-32], [-8], [-28], [-40], [72], [74], [-30], [-12], [-78], [-78], [-36], [38], [-12], [-6], [-14], [72], [-12], [-26], [38], [62], [-8], [30], [38], [76], [-6], [-34], [-36], [38], [-4], [-38], [0], [0], [-74], [-62], [-28], [-76], [32], [-44], [-26], [28], [-8], [66], [10], [4], [42], [-58], [-60], [70], [-12], [26], [-26], [-28], [58], [-36], [4], [-22], [18], [68], [56], [-20], [66], [-40], [14], [-24], [-6], [-36], [-42], [-26], [-4], [-32], [26], [28], [0], [-2], [-66], [60], [-22], [-18], [8], [-42], [-30], [-60], [36], [40], [-54], [30], [-62], [-32], [-8], [16], [92], [30], [78], [26], [-32], [14], [-42], [6], [-8], [66], [60], [-26], [-92], [72], [42], [-84], [2], [30], [8], [-66], [-26], [-48], [4], [-66], [-84], [66], [18], [-92], [-76], [28], [10], [-78], [0], [34], [-44], [72], [56], [-6], [-8], [-66], [-12], [24], [26], [-54], [42], [-38], [46], [8], [-66], [-76], [88], [-40], [-22], [92], [96], [98], [26], [36], [-68], [-48], [46], [-32], [-82], [-4], [-14], [-34], [-90], [40], [38], [6], [-64], [62], [62], [-100], [60], [82], [-22], [-24], [-40], [38], [50], [24], [32], [94], [92], [-66], [22], [-44], [-36], [54], [-14], [-24], [-50], [-24], [8]]
