from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1441, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([0,9]))
pari: [g,chi] = znchar(Mod(199,1441))
Basic properties
Modulus: | \(1441\) | |
Conductor: | \(131\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(26\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{131}(68,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1441.be
\(\chi_{1441}(155,\cdot)\) \(\chi_{1441}(199,\cdot)\) \(\chi_{1441}(210,\cdot)\) \(\chi_{1441}(309,\cdot)\) \(\chi_{1441}(331,\cdot)\) \(\chi_{1441}(485,\cdot)\) \(\chi_{1441}(595,\cdot)\) \(\chi_{1441}(804,\cdot)\) \(\chi_{1441}(837,\cdot)\) \(\chi_{1441}(936,\cdot)\) \(\chi_{1441}(1134,\cdot)\) \(\chi_{1441}(1211,\cdot)\)
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: | Number field defined by a degree 26 polynomial |
Values on generators
\((1311,133)\) → \((1,e\left(\frac{9}{26}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
\( \chi_{ 1441 }(199, a) \) | \(-1\) | \(1\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{8}{13}\right)\) |
sage: chi.jacobi_sum(n)