
# q-expansion of newform 1421.2.a.f, 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
    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 = 1421
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], [2], [3], [0], [6], [1], [0], [-2], [0], [-1], [-2], [-10], [12], [8], [-6], [9], [12], [-2], [-4], [9], [-2], [-10], [3], [6], [-8], [12], [-17], [3], [-7], [-12], [-4], [-6], [12], [19], [3], [-13], [4], [-22], [0], [-15], [0], [-2], [6], [-16], [3], [7], [-28], [-17], [-15], [28], [-21], [15], [-17], [18], [3], [6], [6], [-8], [-7], [-21], [25], [-24], [16], [0], [19], [-18], [8], [14], [-21], [-29], [21], [-6], [-8], [-22], [-10], [9], [-18], [-2], [27], [-14], [-15], [-10], [33], [4], [19], [0], [0], [-1], [-12], [23], [18], [-6], [29], [-18], [-28], [-12], [15], [-30], [-20], [-16], [5], [-30], [-30], [30], [-31], [10], [39], [21], [18], [28], [4], [29], [30], [22], [-40], [24], [40], [39], [-24], [-24], [-5], [26], [-30], [21], [19], [27], [11], [15], [46], [16], [20], [-6], [32], [8], [6], [4], [12], [28], [-24], [18], [43], [-9], [-34], [48], [-20], [0], [46], [6], [22], [-9], [2], [6], [20], [-36], [-34], [42], [-1], [18], [-38], [27], [30], [-21], [-16], [-6], [-3], [48], [17], [-2], [-31], [-48], [-27], [-14], [-27], [-10], [-32], [33], [-25], [-24], [-56], [-50], [8], [12], [-1], [-18], [27], [6], [-4], [46], [14], [-51], [-29], [12], [32], [36], [12], [-21], [32], [11], [15], [36], [-18], [37], [-8], [-2], [-36], [-42], [-32], [24], [-18], [40], [-52], [63], [26], [-12], [12], [-26], [41], [-66], [45], [-18], [2], [31], [60], [56], [-60], [8], [-42], [-12], [28], [45], [-4], [52], [44], [-18], [-53], [-72], [-14], [-24], [51], [48], [60], [-29], [-32], [-40], [42], [39], [25], [-27], [41], [-48], [-46], [45], [-24], [-2], [42], [45], [-58], [-50], [39], [43], [8], [-30], [-38], [-50], [0], [-50], [30], [72], [-22], [-51], [31], [11], [-38], [-64], [-14], [-44], [-6], [11], [-10], [75], [51], [-43], [15], [49], [64], [54], [-64], [-54], [43], [39], [-45], [3], [54], [12], [-46], [45], [16], [42], [9], [40], [70], [-27], [56], [-36], [-40], [53], [-48], [-50], [15], [-25], [51], [-60], [-6], [-1], [3], [-59], [72], [0], [-26], [72], [28], [41], [9], [-40], [-66], [28], [32], [16], [12], [69], [-88], [-66], [1], [-66], [-4], [-33], [-85], [-36], [16], [16], [56], [57], [-42], [-19], [63], [24], [-62], [-10], [6], [-18], [-29], [-13], [-72], [91], [-16], [18], [-24], [-12], [-12], [-48], [83], [63], [12], [60], [70], [74], [-12], [17], [62], [30], [31], [-72], [24], [40], [29], [-66], [78], [-68], [-3], [-17], [78], [-39], [8], [-36], [-20], [-3], [20], [73], [-34], [48], [-64], [39], [-12], [-92], [-24], [41], [-47], [-54], [-31], [-90], [-41], [54], [14], [63], [30], [-5], [50], [-36], [-26], [57], [91], [9], [-15], [-4], [26], [42], [102], [37], [-72], [-90], [81], [52], [-96], [0], [-47], [-57], [-75], [24], [52], [-24]]
