# q-expansion of newform 6975.2.a.bi, downloaded from the LMFDB on 24 June 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 = 6975
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 = [[1, 0, 1], [0, 0, 0], [0, 0, 0], [0, -2, -1], [0, -2, 0], [2, 0, 1], [1, 1, -1], [-2, -2, -2], [5, -1, -1], [-5, -1, 2], [-1, 0, 0], [2, 2, -1], [-2, 2, -4], [0, 2, 4], [7, -7, -3], [3, -1, -5], [9, -1, 2], [-6, 0, -4], [-2, 6, -1], [-1, -1, 0], [2, 6, 1], [3, -5, 3], [-5, 5, 1], [1, 3, 0], [0, -8, 2], [-4, 0, -6], [0, -8, -1], [7, 1, 7], [1, 3, 5], [2, 4, -2], [-8, -4, 2], [3, 5, -8], [-3, 1, 7], [-6, 6, 0], [2, 2, 4], [-3, 1, -5], [2, -2, -4], [4, 4, 5], [20, -4, -2], [6, 0, 2], [6, -10, 2], [2, 0, 0], [-5, 1, 2], [-6, 6, -4], [3, -9, -7], [5, -3, -1], [10, 2, -4], [4, -6, -6], [-9, 9, 3], [-10, 8, -2], [5, 5, -1], [2, 12, 4], [-4, -6, -16], [-10, 6, 6], [-5, -5, 5], [10, 6, 12], [11, 3, 6], [14, -6, 0], [-6, -6, 5], [-6, -8, 2], [-18, 8, 5], [11, 3, 9], [0, 6, -7], [7, -5, -10], [14, -8, -1], [-11, 5, 7], [-3, -7, -11], [6, -2, -9], [12, -20, -6], [-5, 5, -1], [-3, 17, -1], [-11, 3, 4], [-8, 0, -8], [8, 2, -12], [-10, 2, 6], [-1, -11, -15], [-5, -3, -14], [24, 2, -2], [15, 1, 12], [-18, 4, 4], [-1, 3, -2], [9, 11, 3], [3, -9, -2], [10, 10, 9], [-8, 0, -8], [-3, 3, -15], [7, 1, 16], [24, -8, -9], [9, 9, 10], [-14, -12, -2], [-3, 11, 3], [29, 1, 2], [-6, -6, -8], [-6, -6, -8], [12, -8, 4], [7, 1, -3], [17, -17, -14], [-2, -10, -8], [14, -6, -8], [-7, -1, 3], [-20, -4, -7], [3, 7, 9], [5, -17, -15], [11, 3, 4], [14, -2, 4], [8, 16, -8], [-6, 10, -8], [12, 10, -4], [-3, -7, -20], [14, -16, -6], [-24, 4, -9], [18, -4, -9], [-6, 24, 0], [-9, -1, 3], [-8, 8, 12], [-5, 7, -8], [-22, 2, -4], [14, 2, -14], [9, -11, 11], [21, -3, -4], [-30, 4, -2], [16, 2, -9], [-5, -9, 11], [-23, 15, 9], [-4, 16, 6], [-10, 2, 10], [-10, 8, 0], [0, 10, 16], [10, -2, 7], [-2, -8, -16], [5, -7, 7], [10, 6, -12], [10, 10, 12], [0, 2, -9], [-23, 9, 10], [5, -21, -19], [25, -3, -1], [2, -4, 0], [-7, -7, -5], [-7, 15, -14], [-12, 16, 14], [11, -19, -10], [-10, 28, 6], [23, -11, -7], [36, 2, 8], [17, -13, -12], [-14, 10, 14], [30, 0, -6], [-8, -20, 16], [0, 12, 16], [24, 0, 0], [3, 9, -18], [-14, -8, -4], [7, 1, -3], [-30, 12, 11], [-12, 22, 16], [28, -4, -12], [-29, 7, -10], [8, 24, -2], [-11, 11, -4], [-11, -1, 11], [3, -17, -19], [44, -14, -6], [13, 21, -16], [-2, 16, 0], [2, -22, 10], [10, -2, 4], [12, 14, -4], [22, -8, 10], [-17, 3, 1], [24, 2, 10], [0, 2, 12], [15, -1, 12], [14, 10, 32], [44, 4, -6], [11, 19, 0], [-40, 8, -2], [10, -8, 10], [6, -8, -9], [4, 6, 32], [-24, 18, 13], [8, 0, 28], [-28, -4, 6], [17, -11, -7], [11, 17, 23], [9, 9, -8], [-12, 0, 24], [2, -26, -13], [-22, -8, -26], [13, -25, -10], [10, -2, -3], [-23, -1, 7], [-7, 13, -1], [-19, -15, 14], [23, 13, 5], [-13, 7, 15], [48, -6, 0], [8, 2, -10], [-10, 8, -12], [-35, 3, -7], [-22, 4, 20], [34, -2, 8], [-8, 8, 8], [10, -24, -4], [-7, -9, -20], [-17, -9, -13], [28, -12, 6], [-1, -7, 29], [30, -2, 2], [-22, -14, 10], [8, -30, 1], [9, -1, -24], [36, -6, 7], [-13, 13, 9], [-44, 10, 8], [-1, 13, 3], [-36, 10, 13], [-18, 30, 12], [-21, -7, -7], [-4, -18, -16], [4, -14, -22], [20, 16, 10], [-24, 4, 8], [6, 10, 29], [5, -5, -17], [24, -10, -10], [19, -5, 21], [-38, -8, -18], [-10, -10, -4], [33, 9, 14], [-8, 6, 23], [-14, -6, -8], [4, -12, 0], [-27, -1, 20], [0, 10, 18], [8, 4, 28], [26, -16, -22], [16, 2, -2], [-2, -8, 16], [8, -22, -26], [-13, -23, 7], [-32, 0, -4], [16, 10, 12], [-18, 4, -18], [-11, 25, 21], [21, -41, -14], [-4, 18, 14], [50, 4, 4], [10, 2, -8], [-15, 3, 15], [12, -18, -24], [-38, 22, 8], [-4, -24, 10], [-35, 3, 7], [52, -6, -2], [25, -5, 8], [-14, 16, 28], [20, 10, 28], [-31, 1, -17], [-12, 10, -24], [14, -18, 9], [25, 7, 19], [-28, 18, 6], [26, -30, 2], [41, 13, 5], [-46, 30, 14], [54, -12, -4], [34, -14, -18], [18, -4, -1], [2, 28, 12], [-39, -1, 15], [0, 6, -20], [12, -12, -17], [-32, 36, 4], [-36, 10, 16], [-34, 24, -3], [17, -5, 7], [-10, -8, 16], [34, -20, 0], [-10, 34, -8], [-17, 17, 25], [-16, -4, -22], [33, -17, -17], [2, 16, -2], [42, -18, -5], [-18, 14, 30], [-2, 2, -29], [-37, 7, -11], [22, -34, -4], [21, -35, -28], [-8, -8, -8], [23, -15, -15], [5, 5, 9], [-13, -13, 22], [44, -22, -3], [-17, 21, -2], [-41, -13, 7], [7, -25, 13], [-16, 16, -6], [18, 2, -8], [-2, -22, 2], [-13, 7, 5], [60, -12, -4], [23, 25, -21], [-7, 33, -13], [-4, 26, -24], [-7, 11, 7], [42, -24, 2], [-32, 20, 14], [2, 28, 36], [31, 29, -11], [-1, 33, 0], [-6, 8, 38], [30, -26, -12], [13, 35, -19], [26, 8, 32], [-6, 2, 14], [-18, 40, -8], [38, 12, 14], [1, -5, 2], [11, 3, 33], [-16, 6, -11], [-38, 26, 12], [-14, -18, 25], [1, -3, -9], [-16, -14, -20], [-14, -18, 8], [-50, -2, -11], [-14, 38, 6], [1, -7, 31], [54, -12, -16], [-15, -7, 7], [-2, 22, 38], [-29, 37, 9], [22, -6, -18], [-3, -17, 9], [4, -34, -24], [65, -7, -1], [19, 9, 11], [4, -10, 32], [26, 0, 19], [-39, 13, -19], [-15, -15, 28], [20, 12, -32], [-4, -26, -26], [33, -1, -14], [-9, -7, -5], [-10, -22, 4], [62, -18, -12], [39, -29, 5], [-27, -7, 27], [16, 2, 41], [8, 22, -12], [30, 4, -14], [-66, -8, -2], [-9, 11, -29], [-24, 36, 22], [14, -8, -8], [43, 3, 11], [11, 13, 1], [-26, 6, 18], [31, -35, 2], [35, 1, 5], [-5, 21, -2], [26, 0, 7], [20, 0, -13], [-19, -7, 15], [36, 2, 4], [3, -19, 27], [1, -17, 28], [-42, -2, 12], [13, 15, 5], [-42, -2, -4], [-14, -18, 2], [54, 4, 7], [22, -36, -18], [23, -3, -4], [-28, 20, -8], [-28, -8, -14], [0, -8, 13], [-45, 25, 20], [-31, -7, -33], [20, 24, 32], [5, -15, -23], [-24, 4, 0], [-1, 9, 33], [8, 4, 24], [8, 46, -19], [-4, 16, -18], [39, -47, -23], [-10, -32, -24], [3, -17, -43], [1, 1, -22], [10, 8, 25], [47, 23, 4], [32, 14, 30], [-10, 42, 0], [42, -34, -6], [-7, 1, -23], [-3, 17, -4], [36, -14, 0], [27, -17, -7], [-12, -40, 11], [-21, 11, 29], [-7, -39, -6], [4, 8, 6], [-54, -10, 14], [31, 1, 14], [34, -12, 4], [-12, 34, 2], [-24, 6, 19], [-18, 20, 14], [64, -8, -4], [29, 1, 19], [76, -30, -14], [-10, -12, 26], [-24, 10, 8], [76, -26, -7], [-29, -13, 5], [1, 27, -1], [21, -9, 0], [22, -6, -13], [5, 11, -9], [-21, -31, 6], [-68, -8, -16], [65, 13, -7], [15, 17, 35], [29, 13, 28], [-16, -20, 0], [42, -38, -14]]