from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(184, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,0,13]))
pari: [g,chi] = znchar(Mod(159,184))
Basic properties
Modulus: | \(184\) | |
Conductor: | \(92\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{92}(67,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 184.o
\(\chi_{184}(7,\cdot)\) \(\chi_{184}(15,\cdot)\) \(\chi_{184}(63,\cdot)\) \(\chi_{184}(79,\cdot)\) \(\chi_{184}(103,\cdot)\) \(\chi_{184}(111,\cdot)\) \(\chi_{184}(135,\cdot)\) \(\chi_{184}(143,\cdot)\) \(\chi_{184}(159,\cdot)\) \(\chi_{184}(175,\cdot)\)
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: | \(\Q(\zeta_{11})\) |
Fixed field: | \(\Q(\zeta_{92})^+\) |
Values on generators
\((47,93,97)\) → \((-1,1,e\left(\frac{13}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 184 }(159, a) \) | \(1\) | \(1\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{15}{22}\right)\) |
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)