from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(169, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([22]))
pari: [g,chi] = znchar(Mod(92,169))
Basic properties
| Modulus: | \(169\) | |
| Conductor: | \(169\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(13\) | 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 169.g
\(\chi_{169}(14,\cdot)\) \(\chi_{169}(27,\cdot)\) \(\chi_{169}(40,\cdot)\) \(\chi_{169}(53,\cdot)\) \(\chi_{169}(66,\cdot)\) \(\chi_{169}(79,\cdot)\) \(\chi_{169}(92,\cdot)\) \(\chi_{169}(105,\cdot)\) \(\chi_{169}(118,\cdot)\) \(\chi_{169}(131,\cdot)\) \(\chi_{169}(144,\cdot)\) \(\chi_{169}(157,\cdot)\)
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: | 13.13.542800770374370512771595361.1 |
Values on generators
\(2\) → \(e\left(\frac{11}{13}\right)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 169 }(92, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{2}{13}\right)\) |
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)