from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(445, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([0,12]))
pari: [g,chi] = znchar(Mod(186,445))
Basic properties
| Modulus: | \(445\) | |
| Conductor: | \(89\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(11\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{89}(8,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 445.o
\(\chi_{445}(16,\cdot)\) \(\chi_{445}(91,\cdot)\) \(\chi_{445}(121,\cdot)\) \(\chi_{445}(156,\cdot)\) \(\chi_{445}(186,\cdot)\) \(\chi_{445}(256,\cdot)\) \(\chi_{445}(271,\cdot)\) \(\chi_{445}(306,\cdot)\) \(\chi_{445}(331,\cdot)\) \(\chi_{445}(401,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{11})\) |
| Fixed field: | 11.11.31181719929966183601.1 |
Values on generators
\((357,181)\) → \((1,e\left(\frac{6}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 445 }(186, a) \) | \(1\) | \(1\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(1\) | \(e\left(\frac{6}{11}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)