
# q-expansion of newform 1350.3.d.f, downloaded from the LMFDB on 19 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
    R = PolynomialRing(QQ, "x")
    f = R(poly_data)
    K = NumberField(f, "a")
    betas = [K.gens()[0]**i for i in range(len(poly_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 = 1350
weight = 3
poly_data = [2, 0, 1]

# The basis for the coefficient ring is just the power basis
# in the root of the defining polynomial above.
hecke_ring_character_values = [[1001, [-1, 0]], [1027, [1, 0]]]
aps_data = [[0, 1], [0, 0], [0, 0], [-5, 0], [0, -1], [9, 0], [0, -8], [-21, 0], [0, 1], [0, 27], [40, 0], [25, 0], [0, -37], [64, 0], [0, -16], [0, 51], [0, 64], [-97, 0], [131, 0], [0, 63], [-17, 0], [117, 0], [0, -41], [0, 104], [41, 0], [0, 64], [-13, 0], [0, 87], [-8, 0], [0, -27], [-8, 0], [0, 96], [0, -189], [37, 0], [0, 184], [109, 0], [-118, 0], [203, 0], [0, 72], [0, 8], [0, -89], [-127, 0], [0, 72], [271, 0], [0, 224], [-147, 0], [141, 0], [-8, 0], [0, 49], [8, 0], [0, 224], [0, -145], [79, 0], [0, 33], [0, -120], [0, -200], [0, -72], [-221, 0], [88, 0], [0, -144], [32, 0], [0, -123], [486, 0], [0, -48], [-281, 0], [0, -368], [-59, 0], [55, 0], [0, -423], [439, 0], [0, 368], [0, -39], [589, 0], [9, 0], [-157, 0], [0, 200], [0, -424], [-296, 0], [0, -275], [-145, 0], [0, -496], [505, 0], [0, -31], [-32, 0], [504, 0], [0, -168], [0, -48], [-752, 0], [0, 432], [597, 0], [0, 600], [0, 119], [507, 0], [0, 303], [-870, 0], [0, 327], [0, -579], [0, -611], [-163, 0], [697, 0], [-389, 0], [0, 768], [0, 440], [0, -328], [-923, 0], [247, 0], [0, -321], [0, 3], [0, -159], [-736, 0], [437, 0], [-335, 0], [0, 181], [965, 0], [275, 0], [0, 344], [-1152, 0], [0, -489], [0, -248], [0, 352], [-577, 0], [-489, 0], [0, 424], [0, -167], [640, 0], [0, 773], [489, 0], [0, -439], [1080, 0], [248, 0], [-848, 0], [0, -24], [-133, 0], [-1271, 0], [0, 381], [81, 0], [0, -315], [-395, 0], [0, 760], [0, 888], [-752, 0], [0, 352], [-531, 0], [0, -248], [705, 0], [0, 1144], [-1639, 0], [0, -699], [-355, 0], [0, 1032], [-1129, 0], [0, 117], [1227, 0], [0, 168], [1005, 0], [0, 552], [-600, 0], [0, 891], [-1465, 0], [0, -181], [0, 288], [0, -352], [491, 0], [0, -360], [0, -333], [0, 425], [-755, 0], [-24, 0], [-793, 0], [0, -1104], [0, -537], [-984, 0], [0, -568], [-1015, 0], [1027, 0], [0, 1029], [1184, 0], [0, -141], [-533, 0], [-616, 0], [325, 0], [0, 976], [296, 0], [0, -1128], [0, -352], [0, 141], [1327, 0], [-1123, 0], [-631, 0], [0, 1336], [-1617, 0], [0, 856], [-1235, 0], [0, -192], [0, -288], [0, 963], [9, 0], [1857, 0], [0, 651], [0, -895], [0, 456], [-963, 0], [-369, 0], [-407, 0], [0, 761], [0, -957], [-2355, 0], [0, -1385], [0, -485], [1645, 0], [-79, 0], [0, 1176], [965, 0], [0, 1137], [0, -425], [-863, 0], [1970, 0], [0, -797], [0, -1447], [0, 259], [1791, 0], [-674, 0], [0, -240], [-712, 0], [0, 1017], [351, 0], [0, -152], [0, -1793], [-2557, 0], [0, 1976], [-583, 0], [-608, 0], [2600, 0], [0, -429], [19, 0], [0, 1864], [-1065, 0], [0, -112], [0, -897], [0, 456], [0, 903], [1019, 0], [1363, 0], [-377, 0], [0, -776], [0, -975], [645, 0], [0, -312], [1904, 0], [0, -1247], [-936, 0], [0, -64], [0, -352], [745, 0], [0, -1267], [0, -1560], [1711, 0], [1610, 0], [0, -1195], [-736, 0], [1701, 0], [0, -1865], [697, 0], [-97, 0], [0, 312], [1712, 0], [0, 2139], [0, -1549], [784, 0], [0, 2376], [-415, 0], [995, 0], [-2098, 0], [-155, 0], [832, 0], [-307, 0], [0, -1033], [-2344, 0], [2960, 0], [0, 1047], [0, -1041], [-1858, 0], [0, -175], [1945, 0], [-355, 0], [0, 272], [-1425, 0], [0, -688], [-811, 0], [0, -763], [0, -1051], [0, 2073], [0, 1971], [0, -1512], [3640, 0], [0, -1944], [1563, 0], [0, 264], [0, 2615], [165, 0], [1799, 0], [0, 555], [-910, 0]]
