
# q-expansion of newform 1520.2.a.f, downloaded from the LMFDB on 06 September 2025.

# 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 = 1520
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], [0], [4], [-6], [-6], [1], [-8], [-2], [0], [2], [2], [-4], [8], [-6], [4], [-2], [-8], [-8], [2], [8], [-4], [-14], [14], [-10], [-4], [8], [-10], [14], [-20], [-12], [-14], [20], [6], [16], [-10], [-12], [12], [-14], [12], [-2], [-8], [14], [-10], [-16], [-4], [-12], [0], [-26], [-6], [24], [26], [28], [22], [24], [6], [-16], [-2], [18], [-28], [-6], [16], [24], [26], [26], [20], [-2], [20], [-34], [-22], [24], [-8], [10], [-20], [-20], [-26], [6], [-30], [10], [-12], [-10], [8], [-34], [-32], [4], [18], [-22], [30], [16], [12], [0], [28], [28], [-20], [40], [-18], [-22], [-16], [-2], [32], [-42], [24], [42], [-28], [-14], [-44], [-30], [0], [2], [-28], [-2], [18], [4], [-24], [42], [-4], [-8], [6], [-36], [-42], [-26], [-6], [-16], [12], [-34], [6], [16], [-48], [-18], [-28], [-12], [-40], [-10], [10], [-14], [18], [16], [2], [42], [28], [-10], [32], [-48], [-10], [-48], [-26], [-34], [-4], [-36], [50], [-14], [4], [36], [24], [24], [0], [-30], [-46], [22], [-36], [30], [-56], [44], [-58], [36], [56], [-2], [10], [14], [-20], [-58], [-24], [-38], [-32], [-54], [-36], [-10], [36], [-18], [16], [-12], [-30], [-18], [32], [-10], [50], [24], [10], [24], [-10], [36], [4], [46], [-12], [6], [66], [46], [-30], [16], [46], [-24], [50], [26], [52], [6], [-32], [-48], [26], [-36], [2], [-10], [-8], [32], [72], [-14], [8], [50], [-28], [14], [-58], [72], [-6], [16], [-32], [-10], [62], [-48], [4], [-36], [14], [4], [8], [-30], [-44], [-56], [2], [54], [-60], [-16], [56], [76], [-32], [14], [30], [-16], [40], [-52], [60], [24], [14], [2], [-72], [10], [-2], [-12], [70], [-16], [26], [2], [36], [-72], [-26], [-30], [-62], [-68], [30], [-22], [40], [54], [54], [-24], [18], [16], [-2], [0], [12], [-50], [50], [-84], [-4], [-72], [40], [-34], [20], [-48], [50], [2], [-32], [18], [-18], [-12], [-74], [-60], [50], [30], [-16], [6], [-36], [-28], [18], [-62], [64], [-16], [76], [-18], [0], [-34], [48], [-58], [-88], [-42], [-30], [-24], [8], [-46], [76], [72], [66], [18], [-28], [-14], [-82], [12], [34], [-70], [-4], [-32], [12], [-2], [-18], [82], [-80], [-44], [68], [12], [-90], [-58], [10], [16], [-22], [18], [-66], [8], [66], [-4], [-58], [-80], [-16], [-50], [-76], [14], [-18], [44], [-58], [-18], [40], [12], [10], [68], [-98], [26], [52], [-36], [-84], [-10], [-26], [52], [34], [60], [-12], [-8], [78], [-64], [22], [84], [72], [-46], [50], [-10], [-42], [42], [-48], [50], [-100], [-4], [16], [78], [-36], [-60], [18], [-6], [12], [-36], [-32], [94], [-24], [-30], [12], [-42], [-58], [34], [68], [-34], [-66], [-16], [-10], [-30], [104], [-92], [54], [54], [56], [28], [34], [-66], [72], [4], [-6], [36], [86], [-22], [8], [76], [86], [-94], [-8], [10], [-52], [24]]
