from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(323, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([9,22]))
pari: [g,chi] = znchar(Mod(319,323))
Basic properties
| Modulus: | \(323\) | |
| Conductor: | \(323\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | 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)
|
Galois orbit 323.w
\(\chi_{323}(13,\cdot)\) \(\chi_{323}(21,\cdot)\) \(\chi_{323}(72,\cdot)\) \(\chi_{323}(89,\cdot)\) \(\chi_{323}(98,\cdot)\) \(\chi_{323}(166,\cdot)\) \(\chi_{323}(174,\cdot)\) \(\chi_{323}(200,\cdot)\) \(\chi_{323}(242,\cdot)\) \(\chi_{323}(268,\cdot)\) \(\chi_{323}(276,\cdot)\) \(\chi_{323}(319,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
| Fixed field: | 36.0.50089109370343804283025731080360858729207627839973572910891909235301495659033.1 |
Values on generators
\((20,154)\) → \((i,e\left(\frac{11}{18}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 323 }(319, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{1}{12}\right)\) |
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)