# q-expansion of newform 5415.2.a.p, downloaded from the LMFDB on 30 April 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
    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 = 5415
weight = 2
poly_data = [-2, 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, 0], [-1, 0], [2, -1], [0, 1], [-4, -1], [-4, 2], [0, 0], [2, 4], [2, 5], [2, -6], [-4, -5], [6, 1], [2, -1], [6, -4], [-4, 0], [0, -6], [0, 4], [-12, 0], [-4, 6], [-2, 0], [0, 8], [-2, -6], [2, -9], [0, 3], [-2, -2], [-4, 4], [-8, 0], [-6, 2], [-4, -6], [-8, 0], [4, 9], [-14, 0], [4, -2], [2, -4], [2, -14], [-2, -14], [-2, 7], [-10, 0], [-8, 10], [-20, 2], [22, -2], [-8, 7], [8, 5], [-12, -2], [-8, -6], [4, 8], [-12, 4], [2, 12], [4, -4], [-14, 4], [4, -1], [-2, 12], [20, -5], [-4, 2], [-18, -10], [10, -13], [-28, -2], [-6, 12], [2, 9], [2, 3], [12, 14], [12, -14], [-28, -3], [-18, 6], [0, -8], [-10, 2], [-24, -3], [2, 6], [18, 8], [-2, 4], [0, 7], [-18, -1], [0, 7], [-10, -10], [-12, 0], [-2, 12], [30, 4], [-6, -17], [-18, 6], [-8, -19], [26, -8], [-12, 6], [-8, 5], [16, -12], [10, -8], [2, -7], [26, 2], [22, -8], [-18, 7], [22, -4], [0, -27], [0, 2], [12, 7], [-4, -14], [-2, 2], [-14, -13], [6, 13], [-4, 14], [-10, -16], [20, 10], [10, 0], [-14, -20], [-6, 9], [0, 14], [-18, -14], [6, 4], [-18, -8], [-12, 24], [26, 2], [-12, 10], [-6, 22], [-4, 10], [-24, -14], [0, 16], [10, -1], [-18, 17], [14, 22], [0, 30], [20, 0], [-10, -6], [-12, -7], [28, 12], [0, -4], [8, -22], [-6, 4], [-24, -8], [-32, 11], [-6, 1], [10, -8], [-4, -4], [4, 8], [6, 22], [14, 2], [18, -20], [-30, -4], [16, 12], [12, -18], [-8, -2], [38, 8], [4, 8], [26, 18], [10, -9], [-26, -16], [30, 6], [4, -22], [-18, 6], [-4, -24], [-8, 24], [20, 8], [-8, -7], [-22, -18], [14, 23], [8, 0], [-24, -10], [12, 8], [-12, 0], [-26, -18], [18, -26], [2, 11], [10, 2], [20, -10], [-18, -7], [-12, -4], [0, -28], [2, -24], [56, 4], [-42, -6], [26, 20], [-22, -4], [36, 14], [58, 0], [20, 23], [10, 4], [24, 24], [22, -2], [8, 22], [10, 18], [-12, 30], [50, 0], [-54, -3], [-4, -24], [44, 13], [0, 6], [14, -20], [22, 12], [-8, 15], [44, -6], [-30, -2], [-40, -7], [-28, -25], [-2, -36], [-12, 4], [-26, 11], [22, -20], [36, -6], [16, 0], [6, -16], [8, -18], [-54, 8], [22, 23], [-16, 0], [-4, 25], [-18, -16], [-20, -19], [-36, -22], [0, 20], [-26, -12], [46, 6], [-28, 0], [26, -22], [-14, -2], [6, 23], [2, 16], [-32, -2], [-38, 10], [6, -13], [-2, -7], [14, 16], [-8, -14], [-18, -12], [-18, 6], [-38, -9], [30, 17], [-22, 4], [-22, -16], [-16, -18], [12, -10], [4, -20], [-28, -11], [-22, 10], [18, -22], [24, 4], [-14, 23], [10, -11], [38, 0], [-4, 36], [6, -12], [28, 13], [24, 10], [-32, 32], [-28, 20], [30, -33], [14, -38], [12, 28], [28, -23], [46, -19], [-12, 20], [18, 2], [46, 2], [-26, 22], [-22, 4], [26, 10], [-54, 14], [32, -2], [-48, 19], [-12, -36], [24, 8], [-16, -8], [-14, 12], [-12, 0], [-44, -20], [-14, -8], [-24, 5], [-16, -34], [-6, 22], [14, 15], [-50, 18], [4, 12], [-64, 6], [-22, -36], [12, 0], [34, -18], [-16, -2], [-8, -31], [30, 21], [-26, -2], [58, 6], [-10, -30], [16, 23], [-20, -12], [-12, 12], [34, 24], [-46, 0], [30, 9], [44, 17], [18, -16], [36, 28], [16, -12], [-54, 1], [66, 2], [-18, -22], [-56, 8], [4, 50], [-16, 35], [42, -18], [16, -4], [-12, 54], [16, 0], [2, 1], [-34, 16], [-28, -12], [52, -16], [-26, 20], [60, -2], [8, 21], [28, -36], [-10, 20], [20, -11], [-26, 10], [-14, -40], [34, -12], [58, 3], [-20, 14], [-2, 6], [-10, -16], [4, -11], [8, -12], [42, 18], [10, -8], [2, 14], [-2, -26], [30, -9], [-20, -42], [44, 2], [-2, 24], [36, 12], [-60, -14], [42, 20], [-4, 46], [-14, -40], [36, -40], [12, 22], [-54, 18], [-4, -18], [-26, -10], [22, 8], [-24, -20], [-24, -16], [54, -5], [8, 17], [-26, -16], [-2, 7], [-70, 6], [-44, 24], [40, 24], [-8, -60], [0, -28], [-12, -24], [6, -32], [-58, 14], [12, -23], [-30, -22], [12, -10], [30, -26], [-48, 0], [-36, 9], [20, -23], [-48, -10], [-28, 16], [-58, 16], [6, 0], [36, -36], [-40, 34], [14, -41], [16, 3], [10, -20], [-16, -42], [-22, -42], [-24, 37], [-12, 40], [-66, -6], [-26, 3], [-4, -40], [42, -6], [32, 0], [-88, 9], [2, 44], [-26, 50], [-32, 25], [-10, -57], [-8, 14], [2, -39], [-46, -20], [26, -14], [70, -12], [-56, -10], [2, 4], [50, -23], [-58, 20], [18, 26], [4, 4], [24, 45], [66, 13], [20, -38], [28, 31], [-38, -10], [-22, 13], [-2, -30], [-58, -4], [38, -4], [48, -10], [-16, -22], [-60, -6], [-2, -11], [-12, 32], [38, -22], [50, -3], [20, -10], [8, 5], [-12, 11], [52, -6], [-76, -12], [24, 14], [38, -4], [98, 0], [16, -16], [-28, -36], [-2, 24], [20, -16], [-38, -23], [40, 5], [18, -8], [16, -52], [8, -3], [24, -50], [-8, -48], [10, 24], [16, 6], [48, 17]]