from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7935, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,0,19]))
pari: [g,chi] = znchar(Mod(881,7935))
Basic properties
| Modulus: | \(7935\) | |
| Conductor: | \(69\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{69}(53,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7935.s
\(\chi_{7935}(881,\cdot)\) \(\chi_{7935}(1121,\cdot)\) \(\chi_{7935}(1946,\cdot)\) \(\chi_{7935}(2246,\cdot)\) \(\chi_{7935}(3056,\cdot)\) \(\chi_{7935}(3731,\cdot)\) \(\chi_{7935}(5861,\cdot)\) \(\chi_{7935}(6611,\cdot)\) \(\chi_{7935}(7151,\cdot)\) \(\chi_{7935}(7601,\cdot)\)
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: | \(\Q(\zeta_{69})^+\) |
Values on generators
\((5291,4762,7411)\) → \((-1,1,e\left(\frac{19}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 7935 }(881, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{21}{22}\right)\) |
sage: chi.jacobi_sum(n)