
# q-expansion of newform 770.2.i.m, downloaded from the LMFDB on 24 April 2024.

# 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 = 770
weight = 2
poly_data = [49, -217, 982, -75, 381, -43, 113, -6, 12, 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], 1], [[0, 1, 0, 0, 0, 0, 0, 0, 0, 0], 1], [[87558415, -10309236, 46458865, -112651589, 14212810, -1745919, 622151, -114811, 122729, 98501], 383873037], [[1924191734, -108933132, 490910255, -53846440, 150180470, -18448353, 12876532, -1213157, 1296823, -122729], 383873037], [[-10157690340, -27837760897, -8951124856, -7975455071, -1979981394, -1995358828, -215638669, -180540162, -14388479, -12763717], 5374222518], [[16808160187, -32399919421, 11608372179, -10138708261, 4309630396, -2767982449, 566697487, -231347611, 39922932, -21436535], 5374222518], [[2642146213, -11957982735, 149564752, -4666189775, 525637402, -1409087928, 74246393, -149241037, 689507, -12508345], 2687111259], [[-7072023903, -41925208336, -3420594307, -15873003278, -1179137496, -5461553071, -21732244, -636536463, -43187207, -57287134], 5374222518], [[6315431857, -72154008905, -1596430933, -28690819467, 137283730, -7861846037, -4838781, -843704947, -29110928, -63494973], 5374222518], [[13210731065, -59789913675, 3434935019, -23330948875, 2628187010, -7045439640, 371231965, -746205185, 3447535, -62541725], 2687111259]]

hecke_ring_character_values = [[617, [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]], [661, [-1, 0, 0, 0, 0, 0, 1, 0, 0, 0]], [211, [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]]]
aps_data = [[-1, 0, 0, 0, 0, 0, 1, 0, 0, 0], [0, 1, -1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [-1, 0, 0, 1, 1, -1, 0, 0, 0, 0], [0, 1, -1, 0, 0, -1, 0, 0, -1, 0], [-1, -1, 0, 1, 0, 0, 1, 0, 0, -1], [-2, 1, 0, 0, -1, -1, 2, 0, 1, 0], [2, 0, 0, 0, -1, 2, 0, 1, -1, 0], [0, -1, 1, 0, 0, 0, -1, 0, 0, -1], [-2, 1, 0, 0, 1, 1, 2, -1, -2, 0], [2, 0, 1, -2, -2, 3, 0, 1, -1, 0], [2, 0, 1, 0, 0, -1, 0, -1, 1, 0], [-2, -2, 0, 2, 0, 0, 2, 0, 0, -2], [0, 0, 0, 0, -2, 1, 1, 2, 1, -3], [0, 0, 0, 0, -1, -1, -1, 1, -1, -1], [1, -2, 0, -1, 0, 0, -1, -2, -2, 1], [0, 2, -2, 0, 0, 0, -1, 0, 0, 0], [7, 0, -1, -1, 0, 2, 0, 2, -2, 0], [0, -1, 1, 0, -2, 3, 2, 2, 3, -3], [-5, 4, 0, -1, 0, 0, 5, 1, 1, 1], [-4, 0, -1, 0, -2, 1, 0, -1, 1, 0], [1, -1, 0, 1, -2, -2, -1, -1, 1, -1], [-3, 0, -4, -1, 1, -1, 0, 0, 0, 0], [0, -2, 2, 0, 2, -2, -5, -2, -2, 2], [2, 1, 0, -2, 1, 1, -2, 2, 1, 2], [-4, 3, 0, 2, -1, -1, 4, 2, 3, -2], [0, -2, 2, 0, 3, -1, -6, -3, -1, 3], [1, 0, 2, 3, -1, 1, 0, 0, 0, 0], [1, 0, 3, 4, 2, -3, 0, -1, 1, 0], [-2, 3, 0, 0, 1, 1, 2, 1, 0, 0], [0, 0, 0, 0, 1, -1, 3, -1, -1, -3], [2, 0, -3, -1, 3, -2, 0, 1, -1, 0], [-1, 1, 0, -1, -2, -2, 1, -1, 1, 1], [0, 2, -2, 0, -5, 5, 5, 5, 5, -3], [0, 1, -1, 0, 0, -2, 5, 0, -2, 3], [-2, -2, 0, 2, 3, 3, 2, 4, 1, -2], [-12, 0, -3, 0, 2, -2, 0, 0, 0, 0], [-1, 0, 0, -1, -2, -2, 1, 1, 3, 1], [0, -6, 6, 0, 1, -1, -5, -1, -1, -1], [-4, 0, -1, -4, -2, 3, 0, 1, -1, 0], [-4, -5, 0, 5, 1, 1, 4, 1, 0, -5], [0, -2, 2, 0, -4, 3, 1, 4, 3, -3], [5, 0, 4, -1, 3, -3, 0, 0, 0, 0], [0, -2, 2, 0, 0, 3, -5, 0, 3, -1], [11, 0, -1, 1, 2, -4, 0, -2, 2, 0], [-4, 0, -2, 2, -2, 0, 0, -2, 2, 0], [0, -8, 8, 0, 0, 0, -4, 0, 0, -4], [4, 4, 0, 2, 2, 2, -4, 2, 0, -2], [1, 2, 0, 1, 1, 1, -1, 1, 0, -1], [3, 0, -6, 1, -1, -1, 0, -2, 2, 0], [0, -4, 4, 0, -4, 4, 2, 4, 4, -4], [-8, 0, -6, 4, 4, -2, 0, 2, -2, 0], [7, 0, 0, 1, 4, 4, -7, 1, -3, -1], [0, 3, -3, 0, -4, 4, -2, 4, 4, 2], [0, -3, 3, 0, 5, -4, -12, -5, -4, 2], [-7, 6, 0, 3, -4, -4, 7, -1, 3, -3], [0, 6, -6, 0, 1, -3, -9, -1, -3, 3], [18, 0, 2, -4, -6, 6, 0, 0, 0, 0], [0, -2, 2, 0, -2, 4, -6, 2, 4, 2], [4, 0, -6, 2, -4, 4, 0, 0, 0, 0], [-10, 0, -5, 0, 2, -1, 0, 1, -1, 0], [0, -1, 1, 0, -6, 5, -6, 6, 5, -2], [-5, 6, 0, -1, -4, -4, 5, -3, 1, 1], [1, 5, 0, -1, -2, -2, -1, 0, 2, 1], [-9, 0, 0, 1, -4, -4, 9, 3, 7, -1], [8, 0, 3, -4, -7, 8, 0, 1, -1, 0], [0, 7, -7, 0, 3, -2, -2, -3, -2, 4], [-2, 0, 8, 1, -1, -3, 0, -4, 4, 0], [0, 2, -2, 0, -6, 0, 2, 6, 0, -2], [-11, 4, 0, -1, 2, 2, 11, -3, -5, 1], [0, -1, 1, 0, -3, 4, 12, 3, 4, -6], [-5, 6, 0, 5, -6, -6, 5, -3, 3, -5], [-5, 0, -6, -1, 5, -1, 0, 4, -4, 0], [8, 7, 0, -2, -3, -3, -8, -4, -1, 2], [0, -6, 6, 0, -2, 2, 2, 2, 2, 4], [16, -9, 0, 1, 3, 3, -16, -3, -6, -1], [20, -5, 0, 0, 2, 2, -20, 0, -2, 0], [0, -5, 5, 0, 5, -2, -6, -5, -2, 4], [10, 0, 4, 0, -4, 6, 0, 2, -2, 0], [2, 0, 7, 4, -2, -3, 0, -5, 5, 0], [0, 6, -6, 0, -1, -3, 1, 1, -3, 1], [10, 0, 0, -2, 4, -4, 0, 0, 0, 0], [7, 6, 0, -3, 0, 0, -7, 3, 3, 3], [10, -3, 0, -2, -3, -3, -10, -8, -5, 2], [-16, 0, -4, 3, -3, 1, 0, -2, 2, 0], [5, -5, 0, 5, 0, 0, -5, 2, 2, -5], [3, 0, 5, -7, -4, -1, 0, -5, 5, 0], [-2, 0, -7, -4, -4, 5, 0, 1, -1, 0], [10, 0, 0, 0, 6, 6, -10, 1, -5, 0], [0, 0, 0, 0, -1, 3, -25, 1, 3, 1], [0, 2, -2, 0, -2, 2, -8, 2, 2, -6], [9, 0, 2, 5, 1, 0, 0, 1, -1, 0], [-4, -10, 0, -2, 2, 2, 4, 2, 0, 2], [5, 0, 14, -6, -2, 0, 0, -2, 2, 0], [8, 5, 0, -8, -3, -3, -8, 0, 3, 8], [0, 1, -1, 0, 4, -1, -6, -4, -1, 1], [-4, 4, 0, 0, 2, 2, 4, 0, -2, 0], [0, 3, 0, -4, 4, 4, 0, -2, -6, 4], [-14, 0, -1, 0, -2, 1, 0, -1, 1, 0], [0, 4, -4, 0, -2, 0, 8, 2, 0, 0], [0, -5, 5, 0, 5, -2, -8, -5, -2, -2], [12, 12, 0, -6, -2, -2, -12, -2, 0, 6], [0, -5, 5, 0, -2, 7, 16, 2, 7, -2], [0, -2, 2, 0, -5, 1, 11, 5, 1, 1], [7, 0, -11, -3, 2, 3, 0, 5, -5, 0], [2, -10, 0, 4, 2, 2, -2, 0, -2, -4], [0, 3, -3, 0, 2, -3, 26, -2, -3, -2], [6, 0, -6, 6, -2, -2, 0, -4, 4, 0], [1, -1, 0, 9, 0, 0, -1, 5, 5, -9], [0, 8, -8, 0, 8, -8, -10, -8, -8, 4], [-5, 0, 4, -13, -5, 9, 0, 4, -4, 0], [0, -2, 2, 0, 2, -2, 22, -2, -2, -4], [4, 0, -3, -4, -7, 4, 0, -3, 3, 0], [0, 3, -3, 0, 2, 2, 6, -2, 2, 6], [24, 0, -3, -2, -1, 5, 0, 4, -4, 0], [0, -3, 3, 0, 3, 2, -4, -3, 2, 4], [12, -9, 0, -2, 7, 7, -12, 3, -4, 2], [-10, 0, -3, 4, 5, -6, 0, -1, 1, 0], [0, 7, -7, 0, 3, 0, 4, -3, 0, 2], [-10, 0, -1, 2, 9, -4, 0, 5, -5, 0], [-8, 0, 0, 4, -6, -6, 8, 4, 10, -4], [0, 0, 0, 0, -4, 0, 9, 4, 0, -5], [7, 8, 0, -3, 4, 4, -7, 7, 3, 3], [3, 0, 10, -7, -2, 2, 0, 0, 0, 0], [10, 7, 0, -2, 1, 1, -10, 0, -1, 2], [11, 3, 0, 3, -2, -2, -11, -4, -2, -3], [22, 0, 5, -6, -6, 5, 0, -1, 1, 0], [-9, 0, 0, 3, -2, -2, 9, 5, 7, -3], [0, 3, -3, 0, 4, -5, -26, -4, -5, 1], [9, 0, 1, 1, 1, -3, 0, -2, 2, 0], [-12, -3, 0, 4, -3, -3, 12, 9, 12, -4], [-8, 0, -1, 3, 7, -8, 0, -1, 1, 0], [-22, -4, 0, 0, 0, 0, 22, 0, 0, 0], [-6, 0, 2, 4, 2, -2, 0, 0, 0, 0], [0, 1, -1, 0, 2, -5, 4, -2, -5, 6], [0, 7, -7, 0, -1, -4, 4, 1, -4, 6], [-14, 0, 5, 1, 5, -8, 0, -3, 3, 0], [0, -3, 3, 0, -1, 6, 10, 1, 6, 2], [8, 0, 5, -2, -1, 0, 0, -1, 1, 0], [-7, 7, 0, 2, -11, -11, 7, -4, 7, -2], [0, 3, -3, 0, 7, -8, -10, -7, -8, 8], [4, 0, -5, 8, 6, -3, 0, 3, -3, 0], [0, -2, 2, 0, 2, 0, 32, -2, 0, 2], [-10, 0, -3, 7, -5, 8, 0, 3, -3, 0], [0, 0, 2, -6, 2, 6, 0, 8, -8, 0], [0, 3, -3, 0, 4, 4, 9, -4, 4, 1], [11, -4, 0, 3, 0, 0, -11, 1, 1, -3], [-4, 3, 0, 10, -3, -3, 4, 6, 9, -10], [5, 12, 0, -1, -2, -2, -5, -7, -5, 1], [17, 0, 5, -1, -1, 4, 0, 3, -3, 0], [2, 0, 11, 0, -5, 9, 0, 4, -4, 0], [31, -1, 0, -1, 1, 1, -31, -1, -2, 1], [0, 16, -16, 0, -1, -2, -7, 1, -2, 3], [6, 0, 5, 2, -5, 2, 0, -3, 3, 0], [-6, 2, 0, 4, 4, 4, 6, -8, -12, -4], [-6, -2, 0, 2, -3, -3, 6, -1, 2, -2], [-8, 0, -2, 2, -2, 4, 0, 2, -2, 0], [0, -6, 6, 0, -5, 6, 6, 5, 6, -14], [9, -3, 0, -3, 2, 2, -9, -5, -7, 3], [1, 0, 7, -3, 0, -6, 0, -6, 6, 0], [-1, 0, 12, -5, -4, 1, 0, -3, 3, 0], [-15, 0, 0, 3, 2, 2, 15, 9, 7, -3], [0, -16, 16, 0, -2, 8, 24, 2, 8, -2], [0, -1, 1, 0, -5, 4, -12, 5, 4, 4], [0, 5, -5, 0, -4, -1, 4, 4, -1, 5], [0, 0, 0, 0, 2, -2, 26, -2, -2, 6]]
