# q-expansion of newform 2106.2.b.a, downloaded from the LMFDB on 21 September 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 = 2106
weight = 2
poly_data = [18, -24, 16, 2, 2, -2, 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, 0, 0], 1], [[285, 6, -1, -6, 24, -1], 131], [[-147, 292, -136, 101, -11, -5], 393], [[-7, -36, 6, 36, -13, 6], 131], [[-267, 386, 154, 7, -28, 23], 393], [[267, -386, -23, -7, 28, -23], 131]]

hecke_ring_character_values = [[1379, [1, 0, 0, 0, 0, 0]], [1783, [-1, 0, 0, 0, 0, 0]]]
aps_data = [[0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 1], [0, 0, 1, 0, 0, 0], [0, 0, -1, 0, -1, -1], [1, 1, 0, -1, -1, -1], [-1, -2, 0, -1, 0, 0], [0, 0, -2, 0, -2, 1], [2, 1, 0, -1, 0, 0], [1, -1, 0, 1, 0, 0], [0, 0, 1, 0, 3, 0], [0, 0, 0, 0, -2, 4], [0, 0, -3, 0, 2, -1], [0, 2, 0, -2, 0, 0], [0, 0, -1, 0, -1, 2], [6, -3, 0, 0, 0, 0], [0, 0, 0, 0, -4, 5], [0, 1, 0, -4, 0, 0], [0, 0, 2, 0, 4, -2], [0, 0, -3, 0, -6, 0], [0, 0, -2, 0, 2, 2], [-4, -1, 0, 1, 0, 0], [0, 0, -1, 0, 3, 3], [0, 0, -1, 0, -4, -1], [0, 0, -1, 0, 8, 5], [1, -1, 0, -5, 0, 0], [-4, 6, 0, 0, 0, 0], [1, -7, 0, 4, 0, 0], [0, 0, -2, 0, -3, 3], [4, -4, 0, 1, 0, 0], [6, -2, 0, 2, 0, 0], [-4, 4, 0, 2, 0, 0], [0, 0, -4, 0, -2, 4], [-9, -1, 0, -2, 0, 0], [0, 0, 0, 0, -5, 1], [0, 0, 2, 0, 7, -2], [-6, -1, 0, -2, 0, 0], [0, 0, 0, 0, -10, 2], [0, 0, -5, 0, -8, -2], [15, 3, 0, -3, 0, 0], [-6, -6, 0, 0, 0, 0], [0, 5, 0, -2, 0, 0], [0, 0, 0, -6, 0, 0], [0, 0, -1, 0, 10, -5], [0, 0, 2, 0, 12, 0], [2, -4, 0, 4, 0, 0], [17, -1, 0, -2, 0, 0], [0, 0, 4, 0, -9, 6], [0, 0, 4, 0, -4, 5], [0, 0, 0, 0, -22, 2], [3, 6, 0, 0, 0, 0], [0, 0, 4, 0, 11, -4], [0, 0, -1, 0, 6, -3], [-23, -1, 0, -2, 0, 0], [0, 0, 0, 3, 0, 0], [6, 9, 0, -3, 0, 0], [4, 8, 0, -2, 0, 0], [0, 0, -5, 0, -4, -4], [3, -1, 0, -5, 0, 0], [0, 0, -4, 0, 0, 0], [-2, -6, 0, 6, 0, 0], [0, 0, 4, 0, 0, -12], [0, 0, 8, 0, 10, -2], [-2, -4, 0, 4, 0, 0], [-2, 0, 0, -3, 0, 0], [0, 0, 4, 0, 13, -5], [0, 0, -4, 0, 12, 0], [1, -6, 0, -3, 0, 0], [9, 3, 0, -6, 0, 0], [0, 0, 2, 0, -6, -6], [0, 0, 1, 0, 26, -1], [0, 0, 3, 0, -13, -10], [8, -3, 0, 3, 0, 0], [2, 11, 0, -2, 0, 0], [0, 0, 4, 0, 0, -9], [0, 0, -10, 0, -3, 0], [-9, -6, 0, -3, 0, 0], [0, 0, 6, 0, -2, -8], [0, 0, 3, 0, 6, -9], [0, 0, -8, 0, -12, 6], [-1, 13, 0, -4, 0, 0], [0, 0, 6, 0, -17, 7], [0, 0, -4, 0, 25, -2], [4, 12, 0, -3, 0, 0], [-2, 4, 0, 2, 0, 0], [-26, -4, 0, -2, 0, 0], [0, 0, -2, 0, -2, 4], [0, 0, -9, 0, -28, 5], [0, 0, 0, 0, -9, 3], [0, 0, 1, 0, -32, -2], [-3, 3, 0, 12, 0, 0], [0, 0, 4, 0, 7, -8], [0, 0, 5, 0, 2, -10], [10, 2, 0, -8, 0, 0], [0, 0, -5, 0, 17, -7], [6, -9, 0, -3, 0, 0], [0, 0, 4, 0, 22, -2], [21, 6, 0, -12, 0, 0], [9, 5, 0, 10, 0, 0], [0, 0, 4, 0, 21, 3], [0, -10, 0, -2, 0, 0], [0, 0, -8, 0, 10, 10], [7, 11, 0, -2, 0, 0], [5, 4, 0, 11, 0, 0], [9, 1, 0, 2, 0, 0], [0, 0, 4, 0, 2, -10], [0, 0, 5, 0, 13, -8], [0, 0, 9, 0, -2, 1], [24, 3, 0, -3, 0, 0], [16, 12, 0, -9, 0, 0], [-18, -10, 0, 10, 0, 0], [0, 0, -4, 0, -8, 4], [0, 0, 7, 0, -4, 5], [0, 0, 2, 0, 28, -5], [0, 0, -5, 0, -6, -6], [9, 6, 0, 12, 0, 0], [0, 0, 6, 0, -18, 9], [-16, -5, 0, -1, 0, 0], [-22, -8, 0, -4, 0, 0], [-20, -4, 0, 10, 0, 0], [0, 0, 4, 0, 2, 8], [-26, -2, 0, 11, 0, 0], [-18, -3, 0, 0, 0, 0], [0, 0, -1, 0, -1, -4], [0, 0, 5, 0, -15, -3], [23, 7, 0, -13, 0, 0], [0, 0, 4, 0, 10, 4], [-26, -7, 0, 13, 0, 0], [-6, -20, 0, -4, 0, 0], [0, 0, 0, 0, 29, -1], [0, 0, -3, 0, -19, 5], [0, 0, -1, 0, 34, -8], [-22, 0, 0, -6, 0, 0], [9, 7, 0, 5, 0, 0], [0, 0, 6, 0, -24, 0], [0, 0, 2, 0, 44, -4], [0, 0, 0, 0, -12, 18], [0, 0, 1, 0, -25, -7], [20, 1, 0, 8, 0, 0], [-1, 16, 0, -4, 0, 0], [0, 0, 3, 0, -7, 17], [0, 0, -8, 0, 10, 10], [-32, -2, 0, 8, 0, 0], [0, 0, -8, 0, -32, 4], [-1, -7, 0, 1, 0, 0], [0, 0, 1, 0, 23, -4], [0, 0, -2, 0, 29, -7], [-20, -4, 0, -5, 0, 0], [-11, 15, 0, 0, 0, 0], [0, 0, 1, 0, 18, -6], [0, 0, 6, 0, -17, -11], [-3, 12, 0, -9, 0, 0], [-7, -19, 0, 10, 0, 0], [-6, -3, 0, 3, 0, 0], [-2, -6, 0, 12, 0, 0], [-24, 0, 0, 0, 0, 0], [8, -22, 0, 4, 0, 0], [0, 0, -12, 0, -12, 6], [-17, 6, 0, 3, 0, 0], [0, 0, -8, 0, 8, -16], [0, 0, -10, 0, -6, 0], [-12, 6, 0, -12, 0, 0], [0, 0, 7, 0, 14, -16], [15, 3, 0, 6, 0, 0], [0, 0, -17, 0, -8, 7], [0, 0, -6, 0, -15, 12], [6, -13, 0, 7, 0, 0], [-11, -5, 0, 5, 0, 0], [0, 0, -14, 0, -10, 14], [-16, -2, 0, 8, 0, 0], [0, 0, -10, 0, 20, -10], [0, 0, 0, 0, 6, 6], [20, -17, 0, 5, 0, 0], [0, 0, 0, 0, 34, 4], [-8, -17, 0, 11, 0, 0], [3, 6, 0, 0, 0, 0], [0, 0, 3, 0, -17, 4], [0, 0, -4, 0, -14, 22], [4, -11, 0, -7, 0, 0], [-30, 10, 0, 2, 0, 0], [0, 0, -12, 0, 17, 2], [4, 2, 0, 10, 0, 0], [-16, -1, 0, 10, 0, 0], [0, 0, -20, 0, -24, 12], [0, 0, 10, 0, -5, -2], [6, -6, 0, 6, 0, 0], [9, 2, 0, 7, 0, 0], [0, 0, 3, 0, -37, -7], [0, 0, 17, 0, 8, -13], [0, 0, -9, 0, 19, 22], [-3, -10, 0, 1, 0, 0], [0, 0, 7, 0, -21, -3], [17, 21, 0, 6, 0, 0], [0, 0, 0, 0, 11, 5], [-27, -15, 0, 0, 0, 0], [20, -2, 0, 5, 0, 0], [0, 0, -2, 0, 42, 0], [-7, -7, 0, 13, 0, 0], [0, 0, -6, 0, -46, 8], [-6, 0, 0, -6, 0, 0], [0, 0, -12, 0, 12, 0], [-30, -19, 0, -5, 0, 0], [0, 0, -4, 0, 44, -4], [-19, 0, 0, -9, 0, 0], [0, 0, 2, 0, 14, -7], [31, 5, 0, -5, 0, 0], [0, 0, -11, 0, 1, 22], [9, 15, 0, -12, 0, 0], [0, 0, 0, 0, 6, -24], [10, -6, 0, 0, 0, 0], [-21, 4, 0, -4, 0, 0], [-12, -3, 0, 18, 0, 0], [12, 13, 0, 17, 0, 0], [0, 0, 2, 0, -36, 15], [0, 0, 10, 0, -15, -6], [0, 0, -2, 0, 20, -10], [-2, -3, 0, 3, 0, 0], [15, -12, 0, -6, 0, 0], [0, 0, -7, 0, 11, -16], [0, 0, 4, 0, -24, 12], [-8, 22, 0, -4, 0, 0], [0, 0, 6, 0, -9, 18], [0, 0, 10, 0, 52, -2], [0, 0, 6, 0, -1, 14], [27, 3, 0, -12, 0, 0], [32, 3, 0, -6, 0, 0], [5, -8, 0, -1, 0, 0], [-26, -7, 0, 1, 0, 0], [-26, -9, 0, 3, 0, 0], [0, 0, 8, 0, -4, -16], [-34, -6, 0, 18, 0, 0], [42, 8, 0, -14, 0, 0], [0, 0, 0, 0, 41, 2], [9, 12, 0, 6, 0, 0], [-13, 21, 0, -6, 0, 0], [0, 0, -4, 0, 17, 8], [0, 0, -16, 0, -10, 14], [0, 0, -14, 0, 4, -8], [13, 23, 0, -14, 0, 0], [12, 18, 0, 12, 0, 0], [0, 0, 5, 0, 1, -17], [15, 1, 0, 2, 0, 0], [-14, -9, 0, -9, 0, 0], [0, 0, 2, 0, 7, 25], [0, 0, 8, 0, -4, -22], [-2, 5, 0, -11, 0, 0], [0, 0, 1, 0, 33, -18], [0, 0, 1, 0, 17, -13], [0, 0, 9, 0, -23, 7], [-32, 8, 0, -2, 0, 0], [0, 0, 0, 0, 9, -9], [0, 0, -6, 0, 0, -18], [0, 0, -9, 0, -32, -2], [3, 4, 0, -10, 0, 0], [21, 9, 0, 15, 0, 0], [0, 0, 12, 0, 0, -21], [33, 10, 0, -1, 0, 0], [0, 0, -10, 0, -18, 12], [1, -13, 0, 1, 0, 0], [0, 0, 5, 0, 30, -3], [-16, -16, 0, 4, 0, 0], [-45, -15, 0, 6, 0, 0], [0, 0, -14, 0, 14, 14], [-25, -9, 0, 3, 0, 0], [0, 0, -7, 0, 14, -1], [-9, 31, 0, -10, 0, 0], [0, 0, 2, 0, -5, -17], [0, 0, 25, 0, 20, -1], [0, 0, -1, 0, -9, 3], [1, 2, 0, -5, 0, 0], [42, -5, 0, -4, 0, 0], [0, 0, 6, 0, 34, -5], [0, 0, 3, 0, -4, -7], [-22, 15, 0, -3, 0, 0], [10, 0, 0, 9, 0, 0], [0, 0, 11, 0, -29, -2], [0, 0, 10, 0, 24, 9], [0, 0, 6, 0, 43, 1], [0, 0, -1, 0, -12, -9], [-15, 15, 0, 0, 0, 0], [-8, 2, 0, 22, 0, 0], [0, 0, -15, 0, -14, 16], [14, 28, 0, -10, 0, 0], [0, 0, -20, 0, -19, -1], [0, 0, -8, 0, -10, 14], [4, 20, 0, 10, 0, 0], [-13, 14, 0, -2, 0, 0], [0, 0, 0, 0, -16, 2], [0, 0, 7, 0, 8, -28], [-38, -10, 0, 7, 0, 0], [6, 9, 0, 12, 0, 0], [-19, 1, 0, 14, 0, 0], [0, 0, 7, 0, 16, -29], [0, 0, 13, 0, 13, 4], [-14, -2, 0, 14, 0, 0], [19, -7, 0, -11, 0, 0], [10, 4, 0, 14, 0, 0], [3, 33, 0, -3, 0, 0], [13, -19, 0, 10, 0, 0], [0, 0, -10, 0, -10, 32], [-17, -26, 0, 2, 0, 0], [0, 0, -10, 0, -19, -7], [-40, -12, 0, 18, 0, 0]]