from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(316, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([13,24]))
pari: [g,chi] = znchar(Mod(259,316))
Basic properties
| Modulus: | \(316\) | |
| Conductor: | \(316\) | 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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 316.l
\(\chi_{316}(67,\cdot)\) \(\chi_{316}(87,\cdot)\) \(\chi_{316}(131,\cdot)\) \(\chi_{316}(143,\cdot)\) \(\chi_{316}(179,\cdot)\) \(\chi_{316}(223,\cdot)\) \(\chi_{316}(247,\cdot)\) \(\chi_{316}(255,\cdot)\) \(\chi_{316}(259,\cdot)\) \(\chi_{316}(275,\cdot)\) \(\chi_{316}(283,\cdot)\) \(\chi_{316}(299,\cdot)\)
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: | 26.0.234331179334410135333725229627752024799305672723267584.1 |
Values on generators
\((159,161)\) → \((-1,e\left(\frac{12}{13}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 316 }(259, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{11}{13}\right)\) |
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)