from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3204, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,0,20]))
pari: [g,chi] = znchar(Mod(2791,3204))
Basic properties
| Modulus: | \(3204\) | |
| Conductor: | \(356\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{356}(299,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3204.bf
\(\chi_{3204}(91,\cdot)\) \(\chi_{3204}(271,\cdot)\) \(\chi_{3204}(523,\cdot)\) \(\chi_{3204}(631,\cdot)\) \(\chi_{3204}(1135,\cdot)\) \(\chi_{3204}(1351,\cdot)\) \(\chi_{3204}(1819,\cdot)\) \(\chi_{3204}(2359,\cdot)\) \(\chi_{3204}(2467,\cdot)\) \(\chi_{3204}(2791,\cdot)\)
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: | 22.0.4078120343870794568028632139836020814044463104.1 |
Values on generators
\((1603,713,181)\) → \((-1,1,e\left(\frac{10}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 3204 }(2791, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{15}{22}\right)\) |
sage: chi.jacobi_sum(n)