
# q-expansion of newform 845.2.t.f, downloaded from the LMFDB on 06 March 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 = 845
weight = 2
poly_data = [1, 0, 78, 0, 1263, 0, 6176, 0, 11441, 0, 10466, 0, 5353, 0, 1604, 0, 279, 0, 26, 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, 0, 0, 0, 0], 1], [[-409, 2996, 16571, 0, 46015, 0, 68102, 0, 39860, 0, 1214, 0, -7125, 0, -2695, 0, -389, 0, -20, 0], 5992], [[409, 2996, -16571, 0, -46015, 0, -68102, 0, -39860, 0, -1214, 0, 7125, 0, 2695, 0, 389, 0, 20, 0], 5992], [[-539, 0, -19961, 0, -105030, 0, -257858, 0, -324081, 0, -214723, 0, -78322, 0, -15798, 0, -1647, 0, -69, 0], 1712], [[-3773, 39483, -139727, 1183313, -735210, 5800042, -1805006, 9753906, -2268567, 7807469, -1503061, 3425391, -548254, 869910, -110586, 127330, -11529, 9951, -483, 321], 23968], [[26961, -170688, 669321, -1268316, 4331072, -974624, 8524592, 2192820, 7752327, 3934784, 3846937, 2625448, 1108682, 912240, 185192, 174468, 16607, 17360, 617, 700], 11984], [[-2996, -48473, 0, -562582, 0, -2594086, 0, -4719229, 0, -4281808, 0, -2182252, 0, -653341, 0, -113722, 0, -10614, 0, -409], 5992], [[-30699, 71073, -501629, 481307, -1272982, 5445918, -1613590, 13309334, -1440917, 13910023, -870831, 7689981, -319614, 2423698, -66542, 436590, -7171, 41725, -309, 1635], 23968], [[-26961, -170688, -669321, -1268316, -4331072, -974624, -8524592, 2192820, -7752327, 3934784, -3846937, 2625448, -1108682, 912240, -185192, 174468, -16607, 17360, -617, 700], 11984], [[83703, 159919, 3775361, -518719, 23917926, -5625126, 47979462, -10820986, 45500681, -9110155, 23752795, -4108073, 7212374, -1064734, 1266230, -158886, 118807, -12681, 4593, -419], 23968], [[81509, -166551, 861543, -4030089, 4145562, -21497042, 5871486, -40898326, 3110895, -37985157, 414413, -19648615, -198098, -5941306, -83062, -1041138, -11551, -97615, -565, -3773], 23968], [[39483, 154567, 1183313, 4030089, 5800042, 21497042, 9753906, 40898326, 7807469, 37985157, 3425391, 19648615, 869910, 5941306, 127330, 1041138, 9951, 97615, 321, 3773], 23968], [[-39483, 154567, -1183313, 4030089, -5800042, 21497042, -9753906, 40898326, -7807469, 37985157, -3425391, 19648615, -869910, 5941306, -127330, 1041138, -9951, 97615, -321, 3773], 23968], [[-15221, -7127, -798859, -1923401, -5100714, -13384022, -9971998, -27637370, -9214959, -26598933, -4705845, -14010639, -1404094, -4279914, -243222, -754642, -22593, -71035, -867, -2753], 11984], [[-15221, 7127, -798859, 1923401, -5100714, 13384022, -9971998, 27637370, -9214959, 26598933, -4705845, 14010639, -1404094, 4279914, -243222, 754642, -22593, 71035, -867, 2753], 11984], [[1285, -504763, 1254563, -8585045, 10804742, -42433790, 22849098, -78924566, 21535831, -72399061, 10951069, -37111487, 3217222, -11139534, 545958, -1940134, 49605, -180971, 1863, -6965], 23968], [[23947, 419203, 591213, 9271077, 4439554, 50921018, 7462350, 97040830, 5197801, 89206401, 1753171, 45488043, 273770, 13551722, 9562, 2342074, -2093, 216915, -175, 8297], 23968], [[11984, -381235, 5992, -6571905, 0, -31671424, 0, -58320582, 0, -53105203, 0, -27041293, 0, -8068552, 0, -1398012, 0, -129835, 0, -4979], 11984], [[20195, 679767, -127743, 11443805, -735210, 58130790, -1805006, 110068986, -2268567, 102576749, -1503061, 53334711, -548254, 16210930, -110586, 2853942, -11529, 268635, -483, 10417], 23968], [[-15757, -820129, -139727, -19274543, -735210, -100694134, -1805006, -188614114, -2268567, -172338195, -1503061, -87742737, -548254, -26151990, -110586, -4525598, -11529, -419809, -483, -16083], 23968]]

hecke_ring_character_values = [[677, [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0]], [171, [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0]]]
aps_data = [[0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 1, -1, 1, 0, -1, 0, 1, 0, 1, 1, 1, 0, -1, 1, -1, -1, 0, 0], [0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, -1], [0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, -1], [-1, 2, 3, -1, 1, -1, -1, 1, 2, 0, 3, 3, 3, 1, 0, 1, -1, -1, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, -1, -2, -1, 0, -1, 0, -1, -1, 1, -1, -2, -2, -1, 2, -1, 2, 1, 0, 1], [0, -2, -1, 2, 0, 0, 1, 1, 1, 0, -1, 0, -2, 0, 1, 0, 0, 1, 1, -1], [-1, -1, -2, 3, -4, -2, 2, 2, -1, 0, -1, 0, 1, 0, 1, -1, 3, 1, -2, 0], [1, 2, 1, 1, 1, 1, 1, -1, 1, 2, 0, -2, -1, -1, 0, 1, 0, 0, 1, -2], [1, 3, 3, -5, 5, 1, -2, -2, 1, 0, 2, 0, 2, 0, -2, 2, -2, -3, 1, 0], [0, -1, -2, -1, 1, 3, -3, -2, -3, -2, -2, -3, -4, -2, -1, -1, 0, -1, 0, 3], [4, 4, 3, 0, -2, -2, 5, 0, 2, 3, 2, 3, 2, 1, 2, 0, 1, 1, -1, -1], [0, -2, 1, -1, 1, 1, -4, -1, 2, -2, 1, 0, 0, 1, -3, 2, -3, 0, 2, 0], [-2, 2, 6, -4, 7, 1, -4, -2, 5, 0, 2, 1, -1, 0, -4, 4, -6, -3, 3, -2], [0, -1, 1, -1, 3, 0, -1, -1, 2, 0, -1, -1, -3, -2, -1, 2, -3, 0, 2, -3], [2, 1, -3, 0, -3, -1, 4, 2, -2, 2, 0, -2, 2, 1, 3, -1, 4, 1, -2, 0], [0, 2, 2, -1, 2, 2, 0, -3, 0, -1, -1, 1, 0, -2, 0, 1, -1, -2, 1, 0], [1, 1, 0, -1, 3, 2, 0, 0, 1, 1, -1, -8, -2, -2, -2, 1, -2, 0, 1, -3], [2, 2, 1, -2, 6, 5, -5, -3, -2, -3, -3, -5, -2, -1, -3, 0, -1, -1, 1, 2], [0, -2, -2, 0, -2, 3, -4, -3, -3, -3, -3, -5, -5, -2, -1, 0, 0, -1, 1, 4], [4, -5, -4, 1, -3, -4, 7, 2, 0, 2, 1, 0, -1, 0, 2, 0, 0, 2, 0, 1], [1, -2, -3, 4, -1, 4, -1, 0, -4, -2, -3, -1, -2, 0, 0, -2, 0, 0, 0, 1], [-1, 0, 0, -1, 0, -3, 1, 1, 2, 2, 1, -1, 1, 1, 2, 1, 1, 1, -2, -3], [-4, -9, -9, 4, -6, -4, 3, 1, -7, 3, -7, -5, -7, -3, 8, -4, 8, 4, -2, 3], [-3, 2, 6, -7, 8, 1, -3, -3, 3, -2, 2, 2, -1, -1, -2, 3, -6, -4, 3, 0], [0, 5, 6, -5, 3, 0, 1, 0, 6, 0, 3, 0, 3, 3, -5, 2, -2, -2, 0, -1], [1, 3, -2, -4, 0, -7, 4, -1, 0, 4, 3, 2, 8, 1, 5, 0, 3, 1, -3, 2], [-2, 0, -5, 4, -5, 2, 3, 2, -4, 0, -3, 0, 3, -1, 1, -4, 6, 0, -4, 1], [1, -2, -4, 0, -2, -1, 4, -1, -5, -1, -2, -1, 5, -2, 0, -1, 2, 0, -1, 4], [1, -1, -2, -1, 1, 2, -4, -2, -3, -1, -1, 0, 4, -1, 3, -1, 2, -2, 1, 1], [-3, -4, -3, 8, -11, -6, 7, 6, -2, 0, 1, 3, 4, 5, 7, -2, 8, 4, -4, 1], [0, 2, 0, -2, 4, 4, -11, -7, -4, -3, -3, -4, -3, -5, 0, -1, -2, -4, 1, 1], [-2, -4, -1, 4, 0, -3, 1, 3, 2, 0, 0, 2, -4, 0, -2, 0, 2, -1, -1, 0], [3, 4, 6, -2, 1, 3, -3, -2, -3, -3, 1, 5, 2, -2, -2, 2, -5, -3, 1, 0], [1, 0, -5, -4, 4, 1, -5, -2, -3, -2, -1, -4, -1, -4, 0, -2, 0, -1, -1, 3], [4, -1, -2, 3, -8, -4, 6, 3, -2, 0, -2, 3, 6, 1, -2, 0, 5, 4, -4, 2], [-12, 4, 5, 4, -1, 1, -4, 4, 4, -5, 4, 5, 7, 4, -4, 3, -2, -2, 1, -4], [3, -1, 2, 0, 6, 3, 2, -1, 4, 0, -1, -8, -18, 2, -2, 0, -3, 1, 3, -2], [2, -1, -1, -1, -1, -2, 2, 2, 1, 0, 1, -3, 0, -1, 1, 1, 1, -3, 2, -2], [2, -5, -6, -2, 6, 4, -5, -5, -9, -3, -4, -4, -3, -5, 0, -1, 0, 0, -1, 7], [2, -7, -3, 3, 1, 2, 6, 1, 4, 1, -3, -4, -8, 2, -1, 0, 0, 6, 2, -2], [0, -2, 1, 2, -4, -6, 5, 7, 3, 1, 6, 13, 8, 3, 0, 2, -3, 1, -1, 1], [-1, -4, -4, 6, -5, -1, 5, 4, -1, 0, -1, 1, -2, -2, -2, -1, 2, 4, 1, -2], [2, 1, 5, -8, 5, -7, -7, 3, 10, -2, 7, -1, 4, 4, -2, 10, -5, -3, 3, 0], [0, -1, 1, -2, 5, 0, 4, 0, -1, 0, -3, -2, -4, -3, 0, -1, 0, 3, 0, -3], [-2, -7, -4, 3, 3, 5, -1, -2, -2, -2, -2, -6, -9, -3, 4, -4, 1, 3, -1, 0], [-2, 1, 3, 3, -1, 8, 1, 0, 3, -6, -3, 1, 1, 1, -4, -1, -1, -3, 4, -4], [-14, 4, 5, -1, 11, 10, -16, -5, 1, -5, 3, 1, -4, -1, -8, 0, -8, -4, 2, 1], [3, 0, 4, 2, 2, -2, 0, 4, 2, 0, 6, 11, 11, 2, -2, 4, -4, -2, -2, -2], [3, 5, 2, 7, -10, -3, 15, 6, 1, 0, 5, 0, 11, 2, -4, 2, -2, 1, -3, -3], [-4, -2, -4, -6, -1, -9, 2, 2, 1, 2, 4, 7, 9, 3, 5, 0, 2, 1, -1, 2], [1, -3, -6, -2, -1, -1, 2, -5, -7, 2, -5, -11, -7, -7, 8, -5, 6, 4, 1, 3], [4, 5, 1, -2, -4, -1, 4, 0, -4, 1, 2, 0, 9, 1, 4, 3, 1, 2, -1, -1], [4, 3, -1, -1, -5, -2, 4, 2, -2, 0, 0, 0, 6, 2, 5, -3, 4, 4, 1, -3], [0, 3, 1, 1, -5, -4, 11, 4, -1, 2, 3, -3, -1, 2, 3, 1, 3, 3, 0, -4], [6, -1, -13, 6, -5, 0, 6, -2, -15, 2, -9, -8, -8, -7, 10, -7, 10, 5, -4, 7], [4, 1, -2, 1, 0, -2, 3, 2, -5, 6, -4, 1, 0, -4, 3, -2, 1, 2, 0, -2], [-7, 2, 0, 15, -9, -3, 5, 8, 9, 2, 2, 1, -2, 5, 2, 0, 3, 5, -3, -9], [1, 4, -2, 2, -9, 2, 5, 1, -4, 4, 3, 4, 10, 6, 3, -6, 7, 2, -8, 5], [-1, -1, -8, 6, -11, -7, 4, 1, -8, 1, -3, -4, 3, -6, 7, -6, 10, 4, -6, 5], [0, -5, 0, 0, 2, -2, 1, 0, 2, 2, -3, -2, -13, 2, 2, 2, -1, 1, 3, -2], [2, -3, -4, -3, -7, -6, 3, 4, 2, -2, -1, 2, 1, 0, 0, -2, 4, 2, -2, 1], [-3, 8, 7, -4, 9, -2, -1, 2, 2, 2, 3, -1, 0, -3, 5, 0, 0, -2, 0, -5], [2, 8, 7, 2, -4, -15, 13, 10, 13, 14, 9, 3, 8, 4, 8, 6, 4, 7, -1, -11], [-4, 3, 8, 7, -2, -2, 1, 3, 4, 3, -2, 2, -3, -2, 5, 0, 0, 6, 4, -9], [6, 0, -5, -4, -9, -11, 7, 7, 4, 5, 8, 0, 9, 2, 7, 0, 3, 0, -2, -3], [2, -5, 5, -9, 13, 0, -5, -3, 10, 0, 1, -3, -11, 0, -5, 10, -13, 0, 10, -9], [10, -1, 3, -2, 0, -6, 3, -1, 5, -1, 5, -5, -5, 0, 1, -2, -6, 0, 0, 1], [5, -4, 0, -6, -4, -6, 3, 4, 3, 2, 3, 5, 2, 8, 0, 2, 3, -1, 1, 5], [5, 3, -1, -3, -1, -3, -2, -4, -5, 5, 0, -2, -1, -3, 5, -1, -1, 6, -3, 6], [6, 4, 4, -6, 5, 1, 2, -2, 3, 0, 2, 1, 7, 1, 3, 2, -2, -3, 1, 0], [-9, -7, -4, 10, 2, 5, -4, 3, 0, -4, -7, -10, -12, -7, -1, 0, -1, 3, 5, -6], [-3, 19, 18, -12, 12, 6, -6, -4, 14, 3, 12, 2, 9, 8, -8, 6, -12, -9, 3, -6], [2, 1, 2, -1, 3, 0, -5, 0, 9, 1, 4, 5, 1, -1, -11, 6, -10, -5, 3, -3], [0, 11, 7, 1, 6, 7, 7, 0, 6, 7, 0, -13, -6, 0, 0, -1, 0, 0, -1, -14], [-2, -6, -8, 10, -16, -5, 10, 5, -9, 1, -1, 5, -1, 0, 9, -8, 14, 7, -7, 4], [0, 4, 7, -10, 7, 0, 4, 2, 5, 2, 4, -6, -1, 5, 0, 3, 1, -2, 3, 1], [7, -3, -14, -4, -3, 7, -1, -7, -13, -3, -11, -13, -1, -4, 9, -7, 9, 0, -1, 8], [-1, 3, -2, 9, -20, -12, 7, 8, 2, 4, 1, 4, 4, 5, 4, -2, 14, 7, -11, -1], [-2, 7, 10, -6, 12, 12, -7, -3, -1, -1, 4, 4, 1, 1, -4, -1, -12, -8, 3, -3], [11, 16, 10, -12, 9, 4, -3, -3, 8, 0, 9, 8, 16, 2, -11, 8, -13, -12, 4, -1], [-2, -3, -12, -8, 6, 3, -9, -5, -10, -4, 0, -1, 7, -1, -3, -5, 1, -4, -1, 14], [-1, 1, -4, 0, -1, 1, -3, 0, -1, 4, -3, 2, 5, -4, 0, 0, 1, 0, -2, -2], [2, -1, -2, 0, -14, -6, 7, 5, 1, 1, 2, 8, 15, 1, 0, 1, 4, -4, -1, -1], [1, 7, 5, 9, -5, -3, 2, 2, -1, 2, 0, -2, 2, -1, 3, -2, 4, 1, -3, 0], [-2, 0, 3, -11, 9, 8, -20, -4, 5, -9, 5, 2, 7, 3, -13, 9, -9, -2, 5, 4], [0, 2, 15, -8, 7, -3, 0, -4, 5, 1, 1, 2, -3, -3, 0, 5, -1, -1, 4, -1], [3, -3, -2, 11, -13, -7, 13, 2, -3, 7, 0, 2, -1, 3, 10, -8, 5, 8, -8, 1], [1, 11, 11, 7, -3, -5, 12, 4, 3, 4, 4, -8, -8, -6, 10, 0, 0, 5, -5, -10], [-2, -2, -8, -4, 1, 9, -6, -6, -13, -10, -2, 5, 3, -7, -3, -8, 2, -5, -3, 16], [13, 14, 3, -5, -4, -3, 1, -9, -6, 7, 5, -3, 2, -6, 8, -7, 4, 2, -7, 6], [4, 5, 5, -2, -2, -6, 3, 7, 11, -1, 5, 9, -1, 3, -12, 6, -2, -6, 0, -3], [-3, -6, -5, 21, -20, -4, 9, 11, -3, 0, -5, 18, 6, 6, 1, -3, 7, 6, -11, 1], [1, -1, 3, -7, 3, -9, -6, 4, 7, 2, 8, -4, 2, 0, 2, 6, -2, 1, 5, 0], [-10, -20, -5, 2, 3, 1, 2, 6, 2, -1, -8, -11, -15, -1, 3, -3, 2, 6, 5, -2], [6, -12, -14, 14, -15, -2, 19, 2, -18, 3, -14, -15, -8, -1, 18, -14, 17, 15, -5, 3], [7, 7, 8, -21, 18, 10, -13, -7, 6, -3, 2, 3, -8, 1, -8, 6, 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