from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3025, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,16]))
pari: [g,chi] = znchar(Mod(199,3025))
Basic properties
| Modulus: | \(3025\) | |
| Conductor: | \(605\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{605}(199,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3025.bs
\(\chi_{3025}(199,\cdot)\) \(\chi_{3025}(474,\cdot)\) \(\chi_{3025}(749,\cdot)\) \(\chi_{3025}(1024,\cdot)\) \(\chi_{3025}(1299,\cdot)\) \(\chi_{3025}(1849,\cdot)\) \(\chi_{3025}(2124,\cdot)\) \(\chi_{3025}(2399,\cdot)\) \(\chi_{3025}(2674,\cdot)\) \(\chi_{3025}(2949,\cdot)\)
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: | Number field defined by a degree 22 polynomial |
Values on generators
\((727,2301)\) → \((-1,e\left(\frac{8}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 3025 }(199, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{22}\right)\) | \(-1\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(1\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) |
sage: chi.jacobi_sum(n)