
# q-expansion of newform 961.2.g.m, downloaded from the LMFDB on 26 September 2025.

# 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 = 961
weight = 2
poly_data = [1, 0, 44, 0, 410, 0, 956, 0, 979, 0, 511, 0, 140, 0, 19, 0, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 1], [[255, -360, 2839, 2409, 3382, 6431, 691, 5467, -641, 2196, -358, 458, -65, 48, -4, 2], 186], [[-255, -360, -2839, 2409, -3382, 6431, -691, 5467, 641, 2196, 358, 458, 65, 48, 4, 2], 186], [[-463, 439, -7953, 10699, -17926, 21308, -16144, 16835, -7260, 6619, -1732, 1374, -209, 144, -10, 6], 186], [[93, 1125, 0, 7255, 0, 14844, 0, 14865, 0, 7940, 0, 2250, 0, 315, 0, 17], 186], [[463, 439, 7953, 10699, 17926, 21308, 16144, 16835, 7260, 6619, 1732, 1374, 209, 144, 10, 6], 186], [[354, -532, 6074, -10699, 18145, -21308, 21730, -16835, 12516, -6619, 3653, -1374, 517, -144, 28, -6], 186], [[-143, -1347, -412, -6329, 6617, -6857, 14550, -1382, 10969, 1282, 3713, 716, 575, 130, 33, 8], 186], [[354, 532, 6074, 10699, 18145, 21308, 21730, 16835, 12516, 6619, 3653, 1374, 517, 144, 28, 6], 186], [[538, -2510, 5719, -22747, 19734, -47899, 26249, -44466, 16025, -21362, 4827, -5485, 695, -709, 38, -36], 186], [[61, -225, 303, -7000, 1337, -26132, 2430, -35213, 1770, -21738, 587, -6619, 89, -962, 5, -53], 186], [[36, 1788, -926, 27117, -7987, 62350, -13483, 59919, -9222, 29263, -2966, 7575, -445, 983, -25, 50], 186], [[126, 1489, 3765, 15383, 11369, 41009, 14174, 46581, 8581, 26161, 2608, 7535, 380, 1058, 21, 57], 186], [[597, 658, 7093, 14464, 26132, 32677, 35213, 31009, 21738, 15200, 6619, 3982, 962, 524, 53, 27], 186], [[470, 2510, 8467, 22747, 32530, 47899, 44177, 44466, 27451, 21362, 8411, 5485, 1229, 709, 68, 36], 186], [[359, 3724, 2730, 48574, 11336, 121504, 17671, 128230, 12098, 67940, 3942, 18763, 599, 2558, 34, 135], 186]]

hecke_ring_character_values = [[3, [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0]]]
aps_data = [[0, 0, 1, -1, 0, 0, 0, 0, 0, -1, -1, 0, 1, 1, -1, 0], [-1, 1, 0, 1, 1, 0, 2, 0, -1, 0, 1, 0, -2, 0, 0, 0], [1, 0, 0, -1, 1, 0, -1, 1, 1, 1, 0, -1, 0, -2, 1, 0], [0, 0, -2, 0, 0, 1, 0, 0, 1, 0, 1, -2, 0, 0, 0, 0], [2, -2, 0, -1, -2, 0, 0, 0, 3, -1, 0, 0, 3, 1, -1, 0], [1, 0, 0, 0, 0, 1, 1, 0, 1, -1, -1, -2, 0, 1, -2, 1], [-2, -1, 2, 1, 0, -2, 0, -1, 0, 0, -1, 0, 0, 2, 1, 0], [-1, 2, 2, 0, 2, 0, 2, -1, -1, 1, 0, 2, 0, 0, 1, 0], [0, -1, -1, 1, -1, 1, 1, 0, 0, 0, 0, 2, -1, 0, -1, 0], [-2, -1, 1, 1, -2, -2, 0, 0, 1, -1, 1, 1, 1, 1, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, -6, 6, -1, -1, 5, 2, -4, 1, -1, -1, 2, -5, 1], [0, 2, -1, -1, 1, 0, -1, 2, 0, -1, 1, -1, 1, 0, 1, -1], [0, -4, 0, 0, -5, -2, -6, -2, -1, 4, -3, 0, 4, 1, 3, -2], [-2, 4, 1, 1, 1, -1, 2, -2, -2, 0, 0, 3, 1, 1, 3, -4], [-1, -1, 5, -1, -3, 2, 1, -1, -1, 0, -1, 0, 4, 1, -2, 1], [-4, -2, 1, 0, -2, -1, -2, -4, 1, 2, -4, 4, 4, 1, 1, -1], [6, -1, 0, -3, 1, 2, 1, 3, 5, -5, 2, -4, -1, 0, -4, 2], [1, -1, 2, 1, 1, 3, 3, 2, -2, -1, 0, -2, -1, 4, -6, 3], [-2, 1, 2, -1, 3, 3, 4, 0, -3, -1, 0, 0, -2, 1, -2, 2], [3, -3, -1, -5, -6, -2, -10, 0, 6, 2, -5, 0, 8, 0, 2, -5], [1, 2, 0, 2, 7, 4, 3, 4, -1, 2, 4, -4, -5, -4, -2, 4], [-4, 10, 1, -2, 3, 0, 4, 2, -9, -3, 1, 3, -11, 1, 2, -1], [-3, 0, 3, -3, 5, 4, 6, 1, -4, -5, -3, 0, 0, 3, -9, 5], [0, 5, -1, -2, 6, -2, 2, 1, -2, 3, 3, 3, 0, -3, -1, -3], [-4, 3, -7, -1, 1, -5, -5, -2, 2, 5, 1, 4, -2, -6, 9, -6], [1, 1, 0, 1, -6, 0, -1, -6, -2, 1, -3, 1, -3, 0, 1, -1], [-1, 3, -1, 1, -7, -1, -2, -5, -1, 2, -5, 1, 2, 1, 7, -6], [2, -4, -5, -2, -6, 0, -3, 1, 6, 0, 0, -6, 4, -4, 1, -1], [-1, -6, -8, 0, -14, -1, -11, 1, 12, 2, -1, -8, 4, -2, 2, -2], [-9, 3, 3, 3, 7, 0, 9, -5, -14, -3, -2, 9, -6, 4, 0, 3], [-5, -3, -1, 6, -4, -1, -2, -4, -1, 1, -4, 2, -1, 2, 1, -1], [1, 4, -3, -7, 10, -2, 4, 7, 6, -8, 4, -7, -8, 3, -6, 3], [-4, 2, 2, -6, 6, -1, -5, 2, 8, 0, 0, -9, 0, -7, 0, -1], [2, -4, 4, 1, -2, 6, -1, -5, 3, -1, -6, -3, 8, 3, -5, 4], [-4, -1, -5, 5, -3, -6, -1, -7, -5, 6, -1, 2, -1, 1, 9, -6], [-11, 4, 1, 4, 7, -1, -1, -3, -5, -1, -2, 1, -4, 1, 2, 0], [-3, 6, -1, -2, 5, 3, -2, 5, 0, -7, 2, -3, -5, 5, -8, 5], [0, -3, -13, 0, -6, 5, -4, 2, -1, -3, -1, -5, -2, 2, -5, 5], [-10, 3, 1, -4, 6, -6, -3, 4, -6, -2, 1, 5, -3, 3, 4, -2], [-6, 2, -4, 0, 0, -2, -7, 0, 2, -4, 2, -2, -2, 0, 4, -4], [-10, -4, 4, 3, 2, 0, 5, -3, -5, -3, -4, 1, 2, 10, -7, 4], [2, -6, -6, 0, -5, -2, -4, -2, 3, 0, 2, 2, 7, 2, 0, -4], [9, 8, -2, -7, 5, 2, -4, 5, 3, -2, 6, -5, 0, -2, -1, 1], [-1, -2, 6, 4, -5, 0, -3, -2, -5, 6, -2, -8, -4, -4, 6, 0], [-4, 6, 2, -2, 6, -1, 1, 2, -1, -5, 5, 3, -1, -3, -4, 2], [-2, 13, 4, 2, 10, -6, 0, 4, -3, 3, 6, 3, 1, -4, 8, -6], [-3, 7, 3, 0, 13, -1, 8, 0, -11, 1, 1, 5, -1, -2, 1, 0], [2, 11, -2, -11, 12, 0, 0, 14, 2, -7, 7, -4, -10, -4, -4, 0], [-9, 14, 2, 0, 13, -1, 11, -2, -3, -2, 10, 3, -11, -6, 4, -1], [-3, -5, 0, 1, 1, 3, -4, 5, 0, 0, 0, -10, -8, -2, -5, 8], [-8, 5, -2, 7, -4, -2, -2, 4, -8, -3, 7, -5, -8, -1, 8, -1], [5, 2, -1, 2, -3, 4, 8, -3, -3, 3, 2, 10, -7, -1, 1, -1], [-6, 0, 2, 2, 14, 4, 8, -3, -11, -1, -1, 17, 0, 2, -4, -3], [5, -9, -6, -5, -9, 0, -20, 7, 23, -7, -5, -11, 3, 0, -5, 0], [-7, -2, -11, 8, -12, 1, -1, -4, 2, 0, 0, -7, 1, 2, 9, -8], [7, -3, 1, -2, 9, 0, -4, 8, 4, 0, 4, -5, 0, -2, -9, 7], [-1, 4, -3, 6, -3, 7, 2, -1, -2, -4, 1, -1, -4, 3, -8, 3], [10, -2, -6, -4, -10, -5, -5, 2, 15, 5, 0, -4, 0, -5, 3, 0], [17, -3, -4, -5, -7, -2, 2, 3, 5, 7, 1, 0, -1, -7, 2, 0], [7, -9, 3, 1, -11, -4, -8, -5, 5, 2, -1, 9, 17, 1, 1, -2], [-7, -2, 2, 2, 4, 1, -2, 2, 4, 4, 2, -6, 8, -1, 2, 0], [9, -5, 12, -13, 6, 5, -1, 12, 7, -12, -1, -1, 10, 0, -5, 7], [5, -4, 6, 2, -2, 1, 2, -7, -8, 6, -4, 9, 2, 0, 3, -4], [-5, -6, 5, 8, -24, 5, 12, -17, 1, -5, -9, -1, 12, 11, -5, 0], [-2, -9, 4, -4, -1, -3, -12, -1, 2, 4, -2, -2, 12, -5, 2, -2], [-20, 8, -4, -3, 4, 0, 7, 8, -1, -12, 4, -17, -17, 9, 0, 9], [14, -6, -15, -2, -5, -6, -12, 3, 10, 1, 9, -5, 0, -9, 5, -1], [-3, -4, -6, -1, -4, -6, -6, 1, 10, -2, 6, -2, -8, 3, 5, 0], [-10, -2, 1, -1, 13, 7, 16, 3, -5, -15, 2, -7, -2, 13, -15, 13], [8, 8, 10, 0, -2, -3, -1, -1, 3, 5, -1, 13, 5, -3, 8, -3], [6, 10, -3, 1, 5, 3, 5, -1, 5, -1, 9, -2, -2, -10, -4, 2], [9, -3, 8, -7, -1, -11, -5, -4, 3, 7, 1, 11, 19, -1, 5, -12], [0, 7, -5, -7, 10, 3, 8, 12, 12, -19, 8, -10, -10, 0, -15, 8], [6, -4, 2, -10, 4, -7, -3, 4, 12, 3, -4, -3, 7, -6, -1, 1], [19, -13, 8, 5, -4, 0, 12, -1, 3, 3, 2, 9, 11, -2, 3, 3], [24, -11, -15, -9, -16, 4, -15, 3, 25, 2, 3, -12, 8, -4, -1, -8], [10, -9, -1, -6, -11, -2, -3, 5, 13, -3, 2, 6, 6, 0, -4, 1], [15, -13, -14, 6, -23, 24, 1, -6, 3, 0, -6, -7, 6, 0, -9, 0], [-3, 2, 7, -3, -9, 6, 7, -3, -8, -5, -6, 0, 6, 7, -9, 6], [-3, 5, 5, -1, 6, 8, -1, 0, -3, 0, 0, -7, -13, -9, 0, 8], [1, -5, -7, 13, -16, 9, 5, -13, -11, 15, -14, 1, 5, 1, 6, -3], [3, 7, -8, 6, 8, 4, 9, 5, -9, -5, 11, -4, -14, 0, -4, 1], [7, -11, 3, -9, -4, 0, -12, 13, 22, -8, 4, -21, 4, 0, 1, 4], [-18, 2, 10, 0, 7, -6, 9, -6, -5, 0, -10, -3, -1, 3, 3, 4], [6, -2, 4, 4, -18, 3, 3, -8, -1, 5, 1, 2, 16, 3, 1, 0], [18, -3, 1, -9, -2, 0, -2, 1, 5, 0, 0, 9, 6, -6, -4, 4], [-3, -4, 9, -6, 0, -4, 14, -1, -3, -7, 2, -3, -4, 5, 2, 5], [-22, 3, 8, 10, 5, -1, 12, -11, -27, 5, -10, 14, 2, 10, 10, -5], [2, -14, -4, 2, -13, -4, -2, 1, 2, 0, 4, 8, 2, 0, 3, 0], [8, 4, 11, 16, -9, 4, 16, -5, -8, 12, 6, 0, 7, -12, 12, 0], [-6, 17, 15, -1, 22, 4, 12, -1, -20, -5, -1, 13, -9, -4, -4, 3], [2, -8, -9, -9, -2, 17, -10, 8, 16, -13, -2, -19, -8, 5, -14, 12], [-10, 9, -2, -2, 7, -4, -10, -2, 3, 2, -2, 6, -8, -10, 12, -2], [3, 17, 1, -11, 15, 6, 13, 18, -7, -14, 7, -20, -34, -2, -6, 8], [-6, 11, 8, -4, 9, -4, 7, 9, -8, -13, 4, 5, -11, 5, -1, 1], [17, -10, 0, 1, -11, 21, 0, -1, 6, 1, -4, -10, 11, 3, -20, 10], [10, -12, -9, 0, -7, -7, -17, 4, 27, 3, 7, -3, -2, -6, 7, -4], [-13, 17, 12, 7, -12, -4, 6, -6, -5, 1, -2, 8, -3, 2, 12, -1], [0, -18, 5, 7, 2, -1, -6, 4, 6, 6, 4, -6, 5, 1, 2, 5], [29, -15, 14, -7, -15, 4, -6, 2, 23, -3, -8, -8, 29, -3, -3, -10], [20, -9, -7, 1, -22, 6, -2, -13, 16, 6, -6, -3, 22, 0, -1, -6], [13, 17, 2, -1, -1, 2, 7, 3, 0, 1, 7, -2, -11, -12, 9, -5], [20, -18, -13, -1, -22, 13, 16, -8, -5, -13, -1, 14, 0, 8, -13, 7], [-9, 3, 12, 5, 1, -17, -16, -5, -5, 9, 1, 9, 8, -6, 26, -13], [-11, -2, 2, 0, 18, 2, 8, 5, -7, 1, -4, -10, -8, -2, 3, 2], [-16, 13, 6, -4, 22, -15, 17, 2, -24, -11, 0, 13, -20, 11, 0, 0], [11, -16, -18, 4, -32, 9, -15, -6, -2, 3, -15, -14, 12, 15, -7, -3], [5, 0, 10, 10, -2, 0, 8, -12, -20, 13, -6, 28, 12, -3, 5, -8], [-12, 7, 5, 0, 19, -4, 4, 8, -3, -2, 0, -4, -11, -6, 4, -4], [-19, 2, 11, -14, 20, 5, 4, 14, -7, -29, -5, -6, -1, 19, -34, 17], [7, 17, -11, -10, 6, -3, -4, 6, 12, -8, 9, -12, -1, -4, 6, -6], [0, -12, 0, 8, -22, 8, 4, -16, 0, -10, -8, 6, 24, 13, -8, 2], [-22, 14, 2, 5, 18, 2, 18, -5, -28, 7, 2, 21, -18, 0, 0, 2], [0, 12, 4, -3, 22, -4, 0, 2, -14, 5, -4, 18, -2, -3, 5, -6], [0, 5, 0, 4, -6, 0, -3, -6, -9, 1, -3, 14, 14, 3, 17, -14], [-4, -1, 5, -3, -4, 1, -7, 1, -3, 0, -7, 7, -2, 7, -5, 7], [-1, -2, 10, -8, 26, 5, 9, 6, -5, -1, 1, 9, 7, -1, -15, 1], [-17, 14, 8, 5, 12, -10, 26, -2, -26, -13, 1, 11, -17, 13, 1, 0], [-21, 11, -16, -1, 6, 3, -9, 2, -2, -5, -1, -14, -16, 6, -7, 6], [0, 1, -3, -7, 11, 0, 1, 14, 9, -3, 7, -4, -12, -4, -2, -2], [-7, -3, 11, 4, -4, 7, 15, -12, -7, -1, -15, 11, 11, 13, -9, -6], [-15, -6, -18, 11, -20, -2, 4, -13, -2, -11, -2, 0, -3, 19, -6, 0], [7, -10, 10, 9, -6, -10, -10, -5, -8, 6, -6, 5, 6, 0, 7, -6], [7, -6, 4, -8, 3, -7, -9, 2, -11, 1, -9, 4, 9, -3, -5, -4], [-24, 7, -1, 0, 11, -1, 7, -5, -8, -7, 9, -5, -12, 3, -10, -1], [7, -7, 13, 2, -10, 7, -9, 1, 16, 6, -6, -3, 10, 6, -2, -6], [24, -6, 4, -6, -13, 3, -15, 11, 22, 3, -3, -3, 19, -12, -9, -5], [5, 18, -2, -10, 13, -2, 12, 4, -2, -11, 4, 1, -20, 2, -15, 5], [8, 3, 5, 1, -1, 14, 9, -1, 3, -11, -2, -10, 2, -2, -11, -1], [14, -1, 26, -5, 13, 5, 2, 10, -8, 5, -1, 8, 8, 2, -12, 6], [8, 9, -8, -4, 5, 5, -6, 3, -10, 1, 4, -2, -5, 0, 0, 4], [-3, -17, -25, 4, -6, 7, 0, -6, 22, 10, -1, -12, -5, -2, -2, 2], [8, -16, 12, -13, 2, -7, -6, 13, 21, -19, 0, -10, -1, 13, -12, 6], [-6, 3, -14, -2, -14, -2, 8, -1, 9, -9, -1, -8, -3, 2, -8, 6], [9, -16, -7, -7, -6, 3, -13, 2, 19, -2, -3, -12, -3, -3, -10, 5], [-17, 11, -1, 5, 14, -10, -10, 0, 7, 5, 5, -4, -8, -5, 10, 0], [-5, 3, -4, 0, 5, -9, 0, -7, -3, 10, -11, 18, 15, 8, 2, -9], [-4, 6, -14, 11, 13, -11, 6, -11, 0, 17, 12, 11, -13, -6, 17, -5], [7, -10, 5, 13, -6, -7, 1, -7, -1, 11, 6, 11, 20, -2, 7, -4], [24, -4, -36, -2, -33, -1, -18, 1, 20, 2, 6, -15, -7, -9, 8, -7], [23, 13, 6, 2, -7, 1, 10, 7, -21, 0, 0, 23, 1, -13, 7, -6], [-26, -1, 18, 17, 4, -4, 6, -9, -14, 6, -9, 2, 4, 4, 15, 2], [13, -14, -6, 2, -6, 0, -15, 0, 18, 6, 0, -13, 11, -4, -5, 1], [-9, 8, -6, -5, -4, -4, -20, 2, 1, 5, -5, -23, -10, 0, 3, 0], [-3, -12, -17, -14, 11, 1, -16, 10, 3, -9, 6, 4, -9, 9, -12, 0], [13, 1, 0, -23, -2, -16, -21, -5, 21, 0, 0, 11, 18, -3, 1, -17], [-9, -5, -8, 8, -8, -13, -18, -1, -17, 24, -3, 6, -16, -10, 24, -2], [19, 3, 0, -9, 7, -6, 2, 3, -1, 7, 5, 5, 7, -10, 1, 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