from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2166, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([19,15]))
pari: [g,chi] = znchar(Mod(113,2166))
Basic properties
| Modulus: | \(2166\) | |
| Conductor: | \(1083\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(38\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1083}(113,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2166.p
\(\chi_{2166}(113,\cdot)\) \(\chi_{2166}(227,\cdot)\) \(\chi_{2166}(341,\cdot)\) \(\chi_{2166}(455,\cdot)\) \(\chi_{2166}(569,\cdot)\) \(\chi_{2166}(683,\cdot)\) \(\chi_{2166}(797,\cdot)\) \(\chi_{2166}(911,\cdot)\) \(\chi_{2166}(1025,\cdot)\) \(\chi_{2166}(1139,\cdot)\) \(\chi_{2166}(1253,\cdot)\) \(\chi_{2166}(1367,\cdot)\) \(\chi_{2166}(1481,\cdot)\) \(\chi_{2166}(1595,\cdot)\) \(\chi_{2166}(1709,\cdot)\) \(\chi_{2166}(1823,\cdot)\) \(\chi_{2166}(1937,\cdot)\) \(\chi_{2166}(2051,\cdot)\)
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: | \(\Q(\zeta_{19})\) |
| Fixed field: | 38.38.2596071857261356353343069145711378949501369631563408367246132297653804182620484194163230531206681625953.1 |
Values on generators
\((1445,1807)\) → \((-1,e\left(\frac{15}{38}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 2166 }(113, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{5}{38}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{11}{38}\right)\) |
sage: chi.jacobi_sum(n)