from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(265, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([13,11]))
pari: [g,chi] = znchar(Mod(149,265))
Basic properties
| Modulus: | \(265\) | |
| Conductor: | \(265\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(26\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 265.l
\(\chi_{265}(4,\cdot)\) \(\chi_{265}(9,\cdot)\) \(\chi_{265}(29,\cdot)\) \(\chi_{265}(59,\cdot)\) \(\chi_{265}(64,\cdot)\) \(\chi_{265}(144,\cdot)\) \(\chi_{265}(149,\cdot)\) \(\chi_{265}(184,\cdot)\) \(\chi_{265}(199,\cdot)\) \(\chi_{265}(219,\cdot)\) \(\chi_{265}(229,\cdot)\) \(\chi_{265}(249,\cdot)\)
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: | \(\Q(\zeta_{13})\) |
| Fixed field: | 26.26.15613734721204247367361805745180659459790885009765625.1 |
Values on generators
\((107,161)\) → \((-1,e\left(\frac{11}{26}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 265 }(149, a) \) | \(1\) | \(1\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{17}{26}\right)\) |
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)