# q-expansion of newform 1520.2.a.q, downloaded from the LMFDB on 25 May 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 = 1520
weight = 2
poly_data = [2, -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, 0], 1], [[0, 1, 0], 1], [[-3, 0, 1], 1]]

hecke_ring_character_values = None
aps_data = [[0, 0, 0], [0, 1, 0], [-1, 0, 0], [1, -2, 1], [0, 0, -2], [-4, 1, -2], [-1, 0, 1], [-1, 0, 0], [3, 0, -1], [-1, -4, 1], [-2, 0, 2], [-7, 1, -1], [-8, 2, 2], [2, 4, -2], [0, 0, 4], [0, -7, 2], [-5, 4, 3], [-6, 2, 2], [0, 1, -4], [0, 0, 0], [-3, 4, 3], [-8, -2, 0], [4, 2, 0], [-4, 6, -6], [3, -1, -3], [-2, 2, -2], [-3, 3, 1], [4, -5, 0], [-1, 6, 1], [-5, 5, -3], [-5, 3, -5], [8, 0, 4], [7, -4, -3], [-4, 0, 2], [2, 2, -6], [-8, 6, -8], [-16, -2, 2], [-2, -8, 6], [9, 1, 1], [5, 1, 3], [-6, -2, -2], [-6, -8, 4], [3, -6, 7], [-1, -9, 1], [2, 8, -4], [-5, 2, -1], [-5, 8, -1], [-3, -3, -3], [10, -9, 6], [0, -2, -4], [2, 0, 4], [-9, -6, 3], [-2, -6, 0], [-4, 0, -8], [5, -3, -9], [6, 10, -6], [-14, 0, 0], [-1, -2, 5], [-2, -10, 0], [0, 6, -2], [12, -6, -4], [2, 5, 4], [15, -7, 7], [11, 6, 1], [3, 2, 1], [-2, 5, 8], [-23, 0, -3], [-3, 5, 7], [12, -2, 0], [10, -4, 0], [13, -8, -1], [5, -2, -5], [-8, 0, -4], [-2, 5, 8], [9, -2, 1], [11, 5, -1], [-4, 4, -2], [-6, 10, 0], [2, -8, 12], [-10, -8, 4], [2, 12, -10], [-9, 8, -7], [18, 2, -2], [-5, 13, -3], [8, -14, 4], [6, -10, -2], [-22, -8, 0], [19, -10, 5], [-14, 8, -12], [-6, 14, 2], [-30, 6, -2], [-4, -8, -2], [11, -5, -1], [6, -4, 10], [22, -8, 8], [-5, 4, -1], [4, -10, 10], [8, 12, -10], [4, 15, -4], [10, -2, 6], [5, -13, -7], [6, 18, -8], [11, 5, -1], [-8, -12, 2], [10, -16, 0], [21, -2, 3], [0, -6, -4], [-2, 8, 12], [20, 2, 8], [12, -2, -10], [-33, -5, 3], [-2, -14, 8], [4, 2, -2], [-4, -8, 2], [-16, -4, -6], [-6, -22, 12], [-12, 14, -16], [7, 16, -5], [30, -2, 0], [-11, 4, 1], [17, 0, -5], [13, -1, -9], [-34, 5, 0], [15, -15, 11], [-14, -8, 0], [-18, 10, -14], [14, -4, -4], [-25, 6, 1], [9, -6, 1], [20, -4, 10], [-6, 12, 6], [-9, -5, -1], [-36, 2, -8], [-34, -4, 0], [-7, 4, 9], [-9, 0, -1], [6, 9, 0], [-24, 13, -8], [12, -7, 14], [5, -18, 3], [3, -14, -1], [24, 0, -6], [-13, 14, -5], [34, -9, -6], [-33, 6, -7], [2, -14, 2], [-2, -4, 8], [23, 1, 1], [4, -16, 0], [1, 11, 1], [-38, 9, -12], [12, -14, -4], [-6, -4, 6], [31, -7, -5], [0, 3, 12], [-2, 6, 10], [-7, -2, -3], [-31, -10, -1], [-31, 4, -1], [1, 2, -1], [-42, 10, 6], [-33, 1, -7], [22, -10, 10], [20, -16, 0], [-1, -9, 17], [-23, 1, -3], [0, 6, 8], [4, -14, 2], [-2, 18, 4], [12, -2, -6], [27, 6, 7], [17, 2, 15], [8, 4, -18], [-23, 6, -1], [-14, 16, -14], [-8, 10, -4], [-30, 12, 6], [-4, -18, 12], [-9, 7, -17], [12, -6, 8], [29, 12, 1], [-9, 10, 7], [19, -1, 1], [1, -13, -13], [31, -12, -9], [-6, 4, -12], [-16, -25, 10], [-17, -15, 3], [-20, -12, -2], [35, 2, -1], [-29, -7, 5], [-18, 18, 2], [-12, 0, 0], [-19, 12, -13], [-12, 16, 8], [-3, 1, -9], [1, 18, -9], [-16, 0, -6], [-15, -2, -9], [-25, -2, -13], [16, 10, -10], [-22, 10, -14], [-4, -11, 6], [8, 6, -14], [30, -4, 16], [38, -14, 4], [11, 10, -7], [-10, 11, -6], [17, -22, 15], [13, 6, -3], [-25, -6, 1], [-10, 18, -2], [-50, 10, 6], [8, 3, -8], [34, -12, 6], [26, 18, -8], [25, -18, 5], [14, 4, 16], [-9, -21, -1], [2, 20, -16], [32, -6, 10], [10, 24, -14], [-18, -14, -4], [15, 0, 11], [-4, 3, 0], [-2, -8, 20], [-29, 15, 5], [-8, 12, 0], [-1, 1, 3], [22, 24, -14], [8, -12, 22], [9, -6, 13], [-18, 6, 18], [-24, 26, -10], [12, 6, 12], [-1, -10, 11], [12, 2, -8], [6, 0, -8], [-34, 8, -8], [-12, 22, 0], [-11, -13, 13], [-16, 8, -4], [7, 6, 11], [-5, -8, -7], [-37, -13, 5], [-20, 12, 20], [24, 0, 4], [30, -6, -10], [-4, -28, 0], [-11, 14, -7], [12, 12, 2], [3, -6, -11], [7, 0, -13], [16, -12, 22], [-10, -16, 12], [-48, -8, -4], [6, 16, -4], [5, -13, 13], [11, -5, 25], [16, -6, 2], [11, 23, -1], [-5, -11, -1], [28, 6, -4], [-40, -7, -6], [38, 8, -4], [-17, -40, 15], [5, 8, 19], [-35, 8, 9], [-48, 15, -20], [-28, 20, -18], [-7, -2, -17], [32, -21, 8], [21, 8, 7], [21, -18, -9], [-3, 31, -9], [3, 6, 15], [12, 24, -4], [7, -12, 5], [18, 10, -4], [12, -24, -2], [-23, 25, 5], [35, 14, -3], [39, 12, -5], [-15, 30, -5], [-18, -6, 22], [-40, 16, -18], [23, -22, 29], [-5, -1, 1], [-27, 2, 9], [-30, -8, 12], [4, -32, 14], [32, -16, -16], [-11, -21, 7], [-18, -18, 6], [21, 17, 3], [20, 26, -8], [-22, -16, 14], [-6, 8, 0], [-17, -18, 19], [-36, 16, 8], [-20, -2, -22], [40, -17, -14], [23, 22, -15], [36, -11, 12], [10, -16, 14], [-15, 1, 19], [30, -13, 30], [-67, 8, -9], [-20, 4, 4], [-16, -12, 2], [32, 0, 8], [-26, 34, 6], [-4, -12, -2], [7, 41, -13], [-46, 2, -10], [24, 30, -6], [6, 16, 8], [38, -2, 10], [-3, 22, 15], [14, 0, -8], [-17, 2, -9], [-17, -4, -3], [-25, 32, -3], [43, -19, -9], [23, -20, -11], [10, 10, 0], [-41, -8, 15], [0, -5, -24], [-1, 31, -17], [-32, -22, 6], [-6, -6, -14], [-28, 31, 6], [15, -10, 21], [-22, -24, 2], [38, -24, 20], [42, -16, 14], [-9, 2, -23], [-25, 5, -7], [43, 2, 9], [-11, 44, -7], [-52, 7, -18], [3, 28, -23], [-3, 30, -13], [-62, -4, 10], [-24, -23, 18], [-26, 2, -14], [-26, -40, 12], [-28, 19, 20], [16, 0, 0], [-18, 14, -8], [12, -8, -4], [29, -9, 23], [18, -6, -14], [15, -27, 3], [-20, 10, 2], [25, -17, 27], [-5, -14, -17], [56, -4, -4], [15, 16, -15], [-9, 7, 15], [-16, -22, 22], [-84, 2, 0], [49, -21, -3], [1, -10, -19], [-14, -10, -10], [65, 5, 3], [-20, 10, 2], [-13, -35, 15], [16, -22, -2], [-38, -8, 16], [57, -20, -3], [-3, -38, 21], [-15, -34, 7], [-8, 8, 0], [-38, 6, 16], [-9, 6, -21], [-16, -24, 0], [-6, 32, -12], [-35, 30, 15], [39, -29, 1], [-22, 16, -12], [-50, 0, -8], [-21, 2, -17], [46, 12, -16], [39, 12, -1], [-45, 11, -1], [69, -10, -9], [8, 10, -30], [-2, 20, 2], [25, -25, 21], [30, -20, -28], [-64, -1, -6], [-34, 32, 10], [-26, 14, 6], [0, -34, 16], [-33, 19, 25], [52, -10, 20], [-14, 22, 8], [-40, 4, 16], [6, -8, -20], [22, -8, 0], [58, -4, 12], [49, -19, 17], [31, -29, -15], [-26, -4, 0], [-9, 2, 3], [-6, 30, -24], [-76, 8, 10], [8, 23, 8], [-10, 12, -6], [1, 1, -21], [-10, -20, 24], [26, -13, -18], [-52, 24, 10], [1, 16, -17], [-42, -16, -4], [38, -22, -18], [-57, 15, 7], [3, -8, 1], [-57, 13, -5], [-46, -12, -8], [14, -31, 32], [-1, -10, -13], [43, -2, 11], [39, -27, -15], [-4, -1, 22], [38, 11, -10], [25, -8, -3], [-8, 48, -20], [27, -26, 17]]