# q-expansion of newform 2520.2.a.y, downloaded from the LMFDB on 15 June 2026.

# 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, ZZ
    R = PolynomialRing(QQ, "x")
    f = R(poly_data)
    K = NumberField(f, "a")
    betas = [K([c/ZZ(den) for c in num]) for num, den in basis_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 = 2520
weight = 2
poly_data = [-4, -1, 1]

# The entries in the following list give a basis for the
# coefficient ring in terms of a root of the defining polynomial above.
# Each line consists of the coefficients of the numerator, and a denominator.
basis_data  = [[[1, 0], 1], [[-1, 2], 1]]

hecke_ring_character_values = None
aps_data = [[0, 0], [0, 0], [1, 0], [1, 0], [1, 1], [2, 0], [1, 1], [3, 1], [-1, -1], [1, -1], [-2, -2], [3, -1], [0, -2], [3, 1], [-3, -1], [3, 1], [-4, 0], [7, -1], [3, 1], [-6, 0], [4, 2], [8, 0], [0, 0], [4, 2], [6, 0], [-6, 0], [6, -2], [-13, -1], [4, -2], [15, 1], [-4, -4], [-10, 2], [-1, -3], [3, -3], [5, -1], [4, 4], [2, 4], [11, 1], [-5, 1], [4, -2], [-7, -3], [17, 1], [-14, 0], [-6, -4], [3, 1], [6, 2], [-14, 2], [8, 0], [-4, 0], [7, 3], [21, -1], [-10, 0], [-4, -6], [-20, 0], [13, 1], [9, -3], [-16, -2], [2, 2], [-9, -1], [2, 2], [-12, 0], [18, 0], [12, 0], [-8, -4], [10, 0], [-7, 3], [-6, -6], [-34, 0], [-7, 5], [13, 1], [19, -1], [-4, 6], [-4, 4], [-15, -3], [-14, 2], [15, 1], [-9, 1], [-8, -2], [18, 2], [-32, -2], [-16, -4], [-6, 4], [-10, -4], [16, 2], [12, -4], [1, 5], [-2, -6], [0, 6], [-10, 4], [-34, 2], [22, -2], [-4, 0], [-4, -4], [-23, -3], [-6, 2], [13, -5], [-10, 0], [2, 8], [0, 0], [12, -6], [-9, -3], [-27, -1], [24, -4], [22, -2], [-28, 0], [-18, -4], [-16, 4], [5, -3], [2, 8], [26, 0], [8, 0], [-11, 9], [9, -5], [5, 3], [-40, 0], [-8, -4], [-10, 6], [17, 3], [23, 1], [21, 5], [-9, -1], [-6, 4], [-12, 6], [21, 1], [9, -1], [-35, 3], [10, 0], [30, 2], [22, -2], [2, 8], [-10, -2], [-3, -3], [-24, 0], [-33, 3], [-34, 0], [-20, -2], [-18, 4], [-18, 2], [-12, -10], [-16, 0], [11, 5], [-1, -11], [-14, -2], [23, -5], [-21, 7], [6, -10], [0, -10], [-23, -3], [19, 1], [-3, -7], [-9, -1], [-16, -6], [-21, -7], [43, 1], [31, -3], [-46, 0], [36, -4], [2, -4], [8, 2], [-28, -6], [-33, -5], [7, 1], [4, 12], [-22, 6], [-19, 7], [31, 1], [-16, -8], [4, 2], [0, 10], [0, 10], [17, -7], [15, 7], [16, 10], [12, -2], [20, -4], [8, 2], [-20, 0], [-13, 5], [26, -6], [33, -3], [22, -6], [-18, -6], [-27, -7], [17, 1], [-11, 1], [-20, 6], [-25, 3], [12, -4], [-36, -2], [26, -2], [2, -12], [-1, -5], [26, 10], [-50, 4], [11, -1], [-31, -7], [34, -4], [3, 3], [29, 1], [41, -5], [-19, -1], [6, 2], [0, -6], [-14, 4], [-14, 6], [-42, -4], [16, 8], [-21, 7], [38, 2], [-21, 9], [36, -2], [0, 6], [4, 4], [-4, -12], [22, -2], [44, -2], [32, 8], [18, -8], [13, 1], [-5, -11], [-6, 0], [16, 4], [0, 0], [10, -2], [10, -2], [38, 4], [-3, -7], [-16, -2], [6, -2], [-19, 1], [5, 5], [3, -15], [-36, -4], [50, -2], [12, -8], [-51, -5], [-36, 2], [-5, 9], [13, 9], [4, -4], [33, 5], [-15, -1], [16, 0], [30, -4], [27, 3], [56, -4], [38, 6], [32, -4], [6, -2], [-27, 9], [-15, 9], [-50, 4], [-25, -13], [-6, 4], [2, 16], [-21, 7], [-14, -12], [-8, -4], [30, 4], [12, 6], [-2, 2], [37, 1], [-15, -15], [-18, 8], [-5, 15], [-7, -1], [-27, -9], [30, -4], [-27, -1], [-23, -9], [7, -1], [17, 3], [0, 6], [16, 8], [-36, 10], [6, 6], [-53, 7], [42, 0], [10, -12], [-32, -4], [-37, 5], [-24, 0], [9, 7], [-19, -7], [0, -16], [12, -10], [-12, -6], [59, 1], [30, 6], [-2, 0], [37, 7], [42, -2], [-31, -13], [-40, -4], [-13, 3], [12, -2], [-16, 8], [0, -18], [-12, 8], [2, 6], [4, -2], [7, -15], [-28, -4], [7, -5], [-32, 4], [-36, 2], [-27, 9], [-11, 5], [-22, -8], [15, 11], [-1, 1], [-23, 7], [74, 2], [-25, -11], [33, -3], [-66, 0], [26, 14], [14, -8], [-10, -8], [0, -4], [-9, -3], [-18, -8], [8, 10], [8, -16], [1, 15], [-4, 10], [24, 12], [-30, -6], [-37, 7], [-15, 7], [-8, 10], [47, 9], [-52, 0], [2, -6], [-44, 0], [-10, -10], [60, 2], [-9, 15], [-30, -4], [-50, 6], [-39, -11], [21, -9], [26, -12], [-16, 8], [29, 3], [31, -13], [-25, 11], [-39, 11], [-36, -4], [54, -4], [-11, 7], [-14, 16], [47, 5], [40, -8], [-12, 18], [-49, 3], [54, 10], [66, -2], [-5, -19], [-41, 3], [17, 13], [-26, 16], [-29, 15], [-59, -3], [-70, -2], [-28, -14], [0, -10], [-38, -2], [32, -2], [79, -1], [3, -7], [-61, 7], [55, -3], [30, -2], [-1, -1], [22, 18], [-42, 8], [-8, 2], [2, 16], [-30, -4], [72, -6], [7, 9], [60, -4], [-59, 3], [-25, 9], [-5, 21], [32, -8], [78, 4], [3, -15], [-13, -15], [18, -16], [-38, -12], [-43, -3], [-12, 16], [66, -8], [88, -2], [-6, -10], [28, -10], [22, 6], [67, -3], [17, 13], [-13, 5], [-54, -10], [59, -5], [-28, -6], [-42, 2], [-1, -1], [-6, -10], [50, 6], [52, -8], [-16, 18], [-17, 17], [1, -3], [12, 0], [8, 2], [-42, -8], [-10, 0], [-70, -6], [15, -5], [-9, -11], [-9, -7], [20, 10], [-83, -3], [-22, 14], [30, 4], [34, 8], [-43, -15], [-4, 16], [-13, -11], [-14, -14]]