from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5635, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,0,18]))
pari: [g,chi] = znchar(Mod(834,5635))
Basic properties
| Modulus: | \(5635\) | |
| Conductor: | \(115\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{115}(29,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5635.bn
\(\chi_{5635}(834,\cdot)\) \(\chi_{5635}(1324,\cdot)\) \(\chi_{5635}(2059,\cdot)\) \(\chi_{5635}(2304,\cdot)\) \(\chi_{5635}(3039,\cdot)\) \(\chi_{5635}(3284,\cdot)\) \(\chi_{5635}(3774,\cdot)\) \(\chi_{5635}(4264,\cdot)\) \(\chi_{5635}(4754,\cdot)\) \(\chi_{5635}(4999,\cdot)\)
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: | 22.22.83796671451884098775580820361328125.1 |
Values on generators
\((3382,346,2696)\) → \((-1,1,e\left(\frac{9}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 5635 }(834, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{6}{11}\right)\) |
sage: chi.jacobi_sum(n)