
# q-expansion of newform 3276.2.hi.f, downloaded from the LMFDB on 19 January 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 = 3276
weight = 2
poly_data = [25, -5, -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, 0], 1], [[0, 1, 0, 0], 1], [[-5, -4, 4, 1], 20], [[5, 4, 0, -1], 4]]

hecke_ring_character_values = [[1639, [1, 0, 0, 0]], [2549, [1, 0, 0, 0]], [2341, [0, 0, -1, 0]], [2017, [0, 0, 1, 0]]]
aps_data = [[0, 0, 0, 0], [0, 0, 0, 0], [1, 0, -1, -1], [0, 0, 1, -1], [-4, 0, 2, 0], [-3, 0, 4, 0], [4, -1, -1, -1], [-2, 1, -1, 0], [-4, -1, -1, -1], [5, -1, -4, 2], [3, 0, 3, 0], [3, -4, -2, 4], [3, -3, 0, 0], [-1, 2, 7, -1], [5, 0, -3, -1], [1, 3, -4, -6], [4, -3, -5, 3], [0, -3, 3, 6], [1, 0, 2, 5], [3, 0, -3, -3], [3, 1, 4, 0], [-2, 4, -5, -2], [-2, 4, 0, -4], [6, -5, -7, 5], [-2, 0, -1, -4], [-2, 4, 6, -2], [1, -2, 5, 1], [-4, -1, -1, -1], [-1, -3, -4, 0], [-1, 2, 1, -1], [13, 1, -14, -2], [1, -2, -17, 1], [4, -5, -3, 5], [0, 0, -11, 0], [-4, 6, 2, 0], [3, 3, 6, 0], [5, 4, -9, -8], [-6, 1, -5, 0], [5, -3, 2, 0], [-2, -2, 4, 4], [-2, 4, 8, -2], [-17, 1, 1, 1], [-12, -2, 14, 4], [12, 0, -4, 4], [7, -3, 4, 0], [-1, -6, -6, -6], [-17, 0, 17, 0], [-5, -5, -10, 0], [8, 5, -21, -5], [-20, 0, 9, -2], [1, -2, -11, 1], [0, 2, -2, -2], [8, -7, -9, 7], [-1, 2, 25, -1], [-8, 7, 7, 7], [-4, 8, 8, -4], [0, -5, -5, -5], [1, -6, 4, 6], [21, -2, -2, -2], [14, -6, -22, 6], [4, -8, 8, 4], [-5, 0, 1, -3], [5, 5, -15, -5], [1, -5, 4, 10], [5, -10, -1, 5], [-1, 0, -5, -11], [-6, 8, 2, 0], [-10, 0, 0, 0], [-16, -3, -3, -3], [13, -4, 9, 0], [8, 0, -4, 0], [9, 0, -3, 3], [-4, 8, 0, -4], [0, 1, -1, -2], [3, 4, 7, 0], [-10, 6, -4, 0], [-19, 3, 16, -6], [-8, 0, -8, 0], [6, 3, -15, -3], [8, 1, -17, -1], [-9, -5, 14, 10], [-15, 3, 27, -3], [-30, 0, 18, 6], [-2, 4, -15, -2], [-9, 0, 0, 0], [-5, 10, 23, -5], [15, 0, -11, -7], [9, -2, -16, 2], [-1, 0, 1, 1], [-2, -4, 8, 4], [-1, 2, -1, -1], [-16, 0, 6, -4], [7, -4, -10, 4], [-1, 1, 0, -2], [-20, 0, 15, 10], [-5, 10, -11, -5], [20, 1, -41, -1], [17, -1, -16, 2], [3, -4, -4, -4], [31, 0, -11, 9], [35, -1, -1, -1], [-32, 0, 16, 0], [-28, -3, -3, -3], [2, 5, 5, 5], [-3, 0, 3, 0], [-21, 8, -13, 0], [-7, 3, -4, 0], [-13, 0, -1, -15], [1, 1, -2, -2], [13, -2, -11, 4], [-14, 1, 13, -2], [-9, 0, -4, -17], [-9, 0, 9, 9], [3, 11, 14, 0], [6, 0, -5, -4], [-30, -3, -3, -3], [31, 0, -13, 5], [-8, 16, 8, -8], [-6, 1, 1, 1], [3, -6, 5, 3], [33, 0, -20, -7], [-9, -7, 16, 14], [1, -2, -3, 1], [-8, 1, 15, -1], [-10, 1, 19, -1], [32, 2, 2, 2], [-20, 11, -9, 0], [-20, 8, 12, -16], [-35, 5, 5, 5], [-24, 0, 13, 2], [-15, 0, 10, 5], [13, -1, 12, 0], [-31, 2, 2, 2], [15, -3, -12, 6], [-12, -4, -16, 0], [-19, 10, -9, 0], [2, 5, -9, -5], [-11, -6, 28, 6], [-7, 14, -7, -7], [-2, 4, 18, -2], [-1, -5, 7, 5], [-16, -3, 35, 3], [-14, -3, -3, -3], [0, -2, 2, 2], [8, -16, -18, 8], [-19, 0, 13, 7], [5, -17, 7, 17], [-23, -5, 28, 10], [0, 0, -53, 0], [-13, 0, 1, -11], [8, 4, 12, 0], [13, -7, -6, 14], [-4, -8, -8, -8], [14, 1, 1, 1], [3, -6, 18, 3], [-18, 10, 10, 10], [-18, 7, 11, -14], [4, -6, -2, 0], [-21, 1, 1, 1], [13, 7, 20, 0], [-10, 7, 13, -7], [-3, 6, 1, -3], [3, -13, 7, 13], [28, -4, -24, 8], [18, 0, -12, -6], [17, -11, 6, 0], [-1, 2, -14, -1], [-12, 5, 5, 5], [-4, 8, 0, -8], [1, -2, 37, 1], [13, -3, 10, 0], [13, 0, -17, -21], [-24, 0, 24, 0], [-16, -3, 35, 3], [5, 2, -7, -4], [-5, 10, -23, -5], [11, -6, 5, 0], [13, -13, 0, 0], [11, 4, -15, -8], [-24, 5, 5, 5], [-40, 0, 23, 6], [-42, 2, 2, 2], [17, 3, 3, 3], [29, 0, -21, -13], [-2, 0, -2, -6], [-4, 11, 11, 11], [-11, -2, 13, 4], [3, 0, 3, 0], [-7, 14, 0, -14], [16, 9, -41, -9], [28, -7, -21, 14], [-9, 0, 1, -7], [10, -20, -25, 10], [6, 15, -27, -15], [2, 7, 7, 7], [0, 0, 14, 0], [-5, -14, -19, 0], [4, -13, 9, 26], [20, -12, -28, 12], [3, -6, -17, 3], [-12, 11, 13, -11], [-5, 10, 7, -5], [17, -2, -32, 2], [-23, 6, 17, -12], [13, -21, -8, 0], [-8, 16, -18, -8], [-10, 0, -5, -20], [20, 0, -20, 0], [27, -7, 20, 0], [2, 9, 9, 9], [-4, -3, -3, -3], [14, -16, -16, -16], [-49, -4, 53, 8], [-14, -12, -26, 0], [26, 7, -59, -7], [-8, 0, 9, 10], [-3, -4, 7, 8], [-24, -9, -9, -9], [6, -21, 9, 21], [10, 6, -26, -6], [40, -3, -3, -3], [2, 8, -12, -8], [-9, 0, 3, -3], [-10, 5, -5, 0], [-5, 3, 2, -6], [1, -1, -1, -1], [-42, 9, 9, 9], [26, -5, -5, -5], [-2, -11, 13, 22], [5, 0, -7, -9], [-12, 24, -18, -12], [1, -2, -58, 1], [-17, 2, -15, 0], [33, -5, -28, 10], [12, -8, -8, -8], [-27, -7, -34, 0], [0, -17, -17, 0], [38, 3, -79, -3], [5, -10, -31, 5], [-17, -9, 26, 18], [68, 0, -34, 0], [19, 0, 0, 0], [39, -2, -2, -2], [25, -6, -44, 6], [11, 0, -3, 5], [-7, 14, -41, -7], [23, 0, -1, 21], [-26, 0, 18, 10], [-29, 8, 50, -8], [47, 7, -54, -14], [-35, 7, -28, 0], [22, -5, -39, 5], [9, 7, 16, 0], [5, 12, -17, -24], [17, -13, -4, 26], [-6, -22, -28, 0], [-21, -4, -4, -4], [-21, 0, 22, 23], [-32, 10, 10, 10], [4, -8, -4, 0], [41, -6, -35, 12], [55, 1, -56, -2], [-25, -10, -35, 0], [-37, 1, 36, -2], [10, -27, 7, 27], [-12, 0, 12, 0], [-51, 0, 25, -1], [-10, 6, 14, -6], [-21, 0, -1, -23], [-26, 1, 1, 1], [6, -12, -35, 6], [-17, 14, -3, 0], [63, 0, -28, 7], [-46, -3, 49, 6], [8, -16, 49, 8], [2, -3, -1, 3], [10, -8, 2, 0], [11, 6, 17, 0], [8, -17, -9, 0], [0, 14, -14, -28], [14, -28, -4, 14], [49, 0, -26, -3], [-20, 4, 4, 4], [-3, 7, 4, 0], [75, 0, -41, -7], [-5, 10, 19, -5], [-5, -11, 16, 22], [14, -10, -18, 10], [7, -28, 14, 28], [-3, 6, -63, -3], [-16, 32, 18, -16], [8, 1, 1, 1], [8, -7, -9, 7], [47, 0, -27, -7], [-15, 30, 23, -15], [-27, 13, 14, -26], [12, -24, -21, 12], [-23, -9, 32, 18], [4, 7, 7, 7], [23, -30, -7, 0], [-4, 7, -3, -14], [33, 0, -17, -1], [16, -32, 0, 16]]
