# q-expansion of newform 1421.2.a.n, downloaded from the LMFDB on 24 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 = 1421
weight = 2
poly_data = [1, -3, -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], [[-2, -1, 1], 1]]

hecke_ring_character_values = None
aps_data = [[0, -1, 0], [1, 0, 1], [2, -1, -1], [0, 0, 0], [1, 2, 1], [5, 0, 0], [-1, 1, 3], [1, 3, -1], [2, -4, 0], [-1, 0, 0], [2, -1, -2], [-5, 3, 1], [0, 0, 2], [-3, 2, -1], [1, 0, 3], [-4, -3, -1], [3, -1, -5], [8, 2, 2], [3, 5, -5], [-1, 3, -3], [2, 2, -2], [8, -1, 4], [3, -1, -1], [-3, -3, -7], [0, 8, -4], [-6, 6, 2], [9, -7, -5], [5, -5, 1], [-9, 0, -4], [9, -1, -3], [1, 3, -1], [1, -5, -1], [-1, 7, 7], [6, 4, 6], [0, 7, -1], [-3, -1, -3], [16, -8, -2], [2, -7, 2], [-11, -3, -7], [-6, 0, 6], [1, -3, 1], [-4, -1, -3], [9, -5, 1], [-8, 8, 0], [-2, 4, 6], [-4, 8, 10], [14, 1, -4], [1, -7, -3], [10, 4, 8], [-5, 3, 7], [-3, -6, -2], [5, -7, -9], [4, 3, 9], [-9, -2, 7], [13, 0, -6], [-15, 0, 1], [-9, 1, -11], [0, 3, 2], [-10, 0, 0], [13, -2, 2], [8, -14, -8], [10, -12, 2], [-16, 15, 10], [9, 3, -5], [-3, -8, -6], [-2, -8, 6], [-8, -5, 4], [0, 10, 12], [-14, -8, 8], [9, 6, -2], [2, -12, 10], [-18, 9, 0], [9, -13, -1], [3, -14, -4], [17, -17, -5], [7, 1, 9], [0, -16, -2], [6, -3, 5], [-9, 12, 2], [5, 7, -5], [21, -7, 5], [-18, 4, -6], [-13, 7, -5], [-7, -11, 5], [0, 2, -8], [1, 15, 3], [1, 7, -3], [-8, -6, 6], [9, 7, -3], [9, 9, 17], [-24, -5, -12], [-8, 15, -2], [2, -6, -4], [-29, 14, 7], [18, -12, -8], [-8, 3, -4], [15, -4, 14], [-1, 12, 0], [5, 15, 7], [-34, 6, 4], [-1, -11, 3], [-12, -6, -12], [-11, 2, -1], [24, -2, -8], [13, -5, -9], [0, -6, -12], [6, 4, 10], [11, -12, -16], [6, 1, -6], [6, 2, -10], [14, 1, 12], [-13, -16, 10], [-4, -12, 0], [-1, -8, -15], [9, -11, 5], [9, -5, 3], [0, -4, -6], [3, -5, 1], [4, 8, 12], [-6, 11, 16], [-8, -6, 4], [-29, 2, 2], [-7, -1, -7], [16, -12, 6], [-4, -20, 8], [2, -3, -3], [22, -19, -13], [-14, -16, 10], [5, 11, 19], [15, -1, 19], [4, 1, 14], [-3, -13, -21], [-11, 11, 7], [-5, -19, -3], [6, 12, -4], [7, -15, -7], [14, -22, -10], [-11, -3, -3], [-21, 13, 11], [-7, -3, -5], [-1, -7, -11], [24, -5, 1], [25, -17, -5], [10, -11, 12], [-21, 1, 5], [-1, -18, -9], [12, -10, -18], [-19, 4, 8], [12, 9, -6], [14, -4, -8], [-8, 9, -11], [10, -4, 24], [39, -11, -3], [0, 1, 4], [-3, 19, -5], [-3, -2, -21], [-28, 18, 16], [-14, -8, -4], [-6, -12, -14], [-20, 1, -5], [1, -6, 3], [-25, -8, 2], [-18, 15, 2], [9, 19, 17], [-14, -3, 9], [-1, 6, 9], [-10, 12, -8], [9, 5, 1], [-8, -13, -13], [1, 5, 13], [34, 10, -6], [10, 7, -5], [14, 20, -12], [-15, -1, 1], [12, -2, 12], [-10, 20, 10], [-23, 27, 15], [-32, 18, 2], [-5, -4, 7], [28, 2, 22], [-2, 1, 16], [-3, -14, -5], [2, -20, 4], [-27, 8, 0], [-38, -10, 8], [36, -9, -11], [-1, -7, -13], [-25, 17, 1], [34, -12, -2], [50, 6, -2], [4, 7, -15], [16, -3, 8], [-3, -18, -1], [25, -3, -13], [9, -9, -19], [-14, -20, -2], [35, -9, -9], [-8, -18, 8], [14, 9, 13], [-34, 10, 0], [-38, 2, -10], [1, 5, -7], [13, 13, -15], [9, 17, 7], [3, 29, 3], [-16, 15, -11], [-49, 0, 5], [24, 6, 4], [-20, 13, 9], [8, -11, -4], [-21, 11, 15], [-38, -8, -22], [25, -18, -21], [-24, 19, 18], [-13, -11, -7], [-19, -4, -16], [16, -4, 14], [2, -22, 2], [-33, -7, -11], [-11, 11, -1], [-23, 15, 11], [29, 11, 5], [-11, 11, -21], [-2, 13, -12], [-27, -9, -11], [-9, 25, 5], [-20, 0, -8], [-9, 13, 17], [-29, 26, 11], [-18, 18, -16], [-17, -13, -11], [-36, -8, 12], [5, -25, -9], [-12, 16, 2], [2, 0, -14], [9, -1, -1], [-28, -2, 6], [-4, 8, 12], [-38, -10, 14], [-20, -3, 0], [25, 16, -1], [51, -5, 9], [-12, -10, -16], [-8, -4, -6], [8, 22, 14], [-33, 15, -5], [13, 9, 13], [32, -10, 12], [46, 0, 16], [14, 25, -4], [-39, -13, 9], [5, 2, -16], [3, -28, -11], [-24, 8, 16], [-37, 35, 19], [15, -11, -11], [25, 21, 11], [-11, -8, -37], [22, 16, 10], [42, -19, -13], [-25, 5, -5], [-17, 12, 29], [6, -28, 18], [35, 9, 1], [11, 9, 15], [-32, -17, 6], [8, -4, -24], [-28, 14, -4], [5, 27, 21], [45, 2, -10], [-51, -6, -20], [32, -2, -14], [10, 1, -1], [11, 17, 37], [-41, -17, 1], [-14, -13, 28], [-43, 3, -3], [-11, -20, -30], [11, -13, -15], [-31, 21, 9], [48, -18, -10], [-2, 28, -20], [-24, 8, 14], [-6, -12, -16], [13, -5, -3], [-1, 1, -31], [2, -8, 12], [-13, -13, -7], [13, -9, -13], [24, -1, -13], [-27, -24, 4], [46, 16, 6], [-5, 22, 36], [-33, 34, 23], [8, 18, -16], [-59, 20, 2], [-1, 26, -21], [-5, -28, 18], [6, -2, -8], [15, -23, -7], [42, 16, -16], [-10, -8, -18], [-9, -31, -11], [-21, -12, 1], [21, -1, -15], [-14, -7, -27], [12, -5, 20], [11, -6, -8], [-3, 15, 7], [-15, 12, -6], [0, 22, 10], [36, -8, -10], [-26, 16, 10], [-26, -4, 10], [59, -7, 5], [6, -9, 23], [57, 7, 19], [7, -1, 11], [11, -14, 16], [-33, 21, 21], [-5, 14, 25], [14, -3, 37], [4, -30, -2], [33, -2, 5], [-21, 4, -10], [43, -37, -15], [15, 27, 11], [37, 15, -15], [-58, -3, -4], [-7, -38, 2], [-44, 16, -14], [10, -7, -5], [55, 5, 5], [23, -23, -23], [-13, -6, -41], [-34, 2, 4], [8, -6, 16], [-17, -15, 19], [36, -14, -16], [-2, -22, -36], [-30, 8, 10], [-12, 14, 28], [-64, -2, -4], [-25, -7, -13], [38, -4, 0], [9, 14, -31], [-35, 23, 29], [33, -9, -9], [17, 6, -29], [-36, -2, -6], [63, -17, -3], [-10, 4, -30], [-6, 12, 20], [17, -31, -5], [18, -4, -14], [14, 0, -36], [30, -25, -14], [-1, -7, -17], [-9, 9, 29], [-16, -18, -6], [-5, -30, -26], [-56, -9, 9], [6, -23, -10], [-2, -4, -12], [38, -7, -16], [-2, -26, -20], [-28, -12, 10], [-9, 7, 37], [19, 9, 5], [30, -15, 4], [37, -11, -3], [63, 1, 1], [11, 29, 9], [27, -17, 1], [1, -18, 22], [-37, -7, 9], [-9, 8, -1], [8, -14, 20], [-39, -14, 8], [1, 0, -12], [30, -2, -26], [-8, 19, 13], [-24, 33, 18], [-32, 26, 0], [21, -17, -27], [-8, -12, -38], [-11, 3, 11], [-5, -26, -34], [28, 3, 16], [-61, -7, -15], [35, 21, 33], [11, -16, 22], [-32, 21, -4], [46, 19, 22], [23, -14, -11], [-32, -5, 11], [-23, 15, 7], [53, 5, 9], [30, -44, -22], [41, -41, -13], [-70, -8, 10], [9, 23, -17], [15, 9, -17], [34, 17, -11], [1, 32, 16], [-29, -9, -7], [-43, 22, 4], [37, -29, -13], [-42, -7, 6], [-30, -10, 22], [-12, -6, 12], [-68, 13, 13], [2, -14, 16], [19, 11, 23], [-40, 24, 24], [-11, -9, -15], [23, 16, 41], [4, -2, -18], [-31, 45, 29], [5, 2, 25], [-16, 18, -12], [3, -31, 17], [5, 38, -5], [8, -27, -44], [52, -19, -1], [28, -19, -3], [-60, 26, 2], [1, 25, -3], [24, 2, -20], [44, -41, -26]]