sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(14157, base_ring=CyclotomicField(660))
M = H._module
chi = DirichletCharacter(H, M([220,108,275]))
pari:[g,chi] = znchar(Mod(58,14157))
Modulus: | \(14157\) | |
Conductor: | \(14157\) |
sage:chi.conductor()
pari:znconreyconductor(g,chi)
|
Order: | \(660\) |
sage:chi.multiplicative_order()
pari:charorder(g,chi)
|
Real: | no |
Primitive: | yes |
sage:chi.is_primitive()
pari:#znconreyconductor(g,chi)==1
|
Minimal: | yes |
Parity: | odd |
sage:chi.is_odd()
pari:zncharisodd(g,chi)
|
\(\chi_{14157}(58,\cdot)\)
\(\chi_{14157}(115,\cdot)\)
\(\chi_{14157}(223,\cdot)\)
\(\chi_{14157}(466,\cdot)\)
\(\chi_{14157}(526,\cdot)\)
\(\chi_{14157}(553,\cdot)\)
\(\chi_{14157}(643,\cdot)\)
\(\chi_{14157}(691,\cdot)\)
\(\chi_{14157}(808,\cdot)\)
\(\chi_{14157}(817,\cdot)\)
\(\chi_{14157}(994,\cdot)\)
\(\chi_{14157}(1021,\cdot)\)
\(\chi_{14157}(1138,\cdot)\)
\(\chi_{14157}(1159,\cdot)\)
\(\chi_{14157}(1285,\cdot)\)
\(\chi_{14157}(1345,\cdot)\)
\(\chi_{14157}(1402,\cdot)\)
\(\chi_{14157}(1489,\cdot)\)
\(\chi_{14157}(1510,\cdot)\)
\(\chi_{14157}(1753,\cdot)\)
\(\chi_{14157}(1813,\cdot)\)
\(\chi_{14157}(1840,\cdot)\)
\(\chi_{14157}(1930,\cdot)\)
\(\chi_{14157}(1978,\cdot)\)
\(\chi_{14157}(2095,\cdot)\)
\(\chi_{14157}(2104,\cdot)\)
\(\chi_{14157}(2281,\cdot)\)
\(\chi_{14157}(2425,\cdot)\)
\(\chi_{14157}(2446,\cdot)\)
\(\chi_{14157}(2572,\cdot)\)
...
sage:chi.galois_orbit()
pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\((6293,3511,4357)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{9}{55}\right),e\left(\frac{5}{12}\right))\)
\(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(14\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 14157 }(58, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{201}{220}\right)\) | \(e\left(\frac{91}{110}\right)\) | \(e\left(\frac{347}{660}\right)\) | \(e\left(\frac{41}{660}\right)\) | \(e\left(\frac{163}{220}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{161}{165}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{281}{330}\right)\) | \(e\left(\frac{439}{660}\right)\) |
sage:chi.jacobi_sum(n)