from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7360, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,0,0,19]))
pari: [g,chi] = znchar(Mod(191,7360))
Basic properties
| Modulus: | \(7360\) | |
| 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 \(\chi_{92}(7,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7360.dc
\(\chi_{7360}(191,\cdot)\) \(\chi_{7360}(511,\cdot)\) \(\chi_{7360}(1791,\cdot)\) \(\chi_{7360}(2751,\cdot)\) \(\chi_{7360}(3391,\cdot)\) \(\chi_{7360}(5311,\cdot)\) \(\chi_{7360}(5631,\cdot)\) \(\chi_{7360}(5951,\cdot)\) \(\chi_{7360}(6271,\cdot)\) \(\chi_{7360}(6911,\cdot)\)
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
\((1151,5061,4417,6721)\) → \((-1,1,1,e\left(\frac{19}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
| \( \chi_{ 7360 }(191, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{6}{11}\right)\) |
sage: chi.jacobi_sum(n)