from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6003, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,15,0]))
pari: [g,chi] = znchar(Mod(755,6003))
Basic properties
| Modulus: | \(6003\) | |
| 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 \(\chi_{69}(65,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6003.bm
\(\chi_{6003}(494,\cdot)\) \(\chi_{6003}(755,\cdot)\) \(\chi_{6003}(1538,\cdot)\) \(\chi_{6003}(1799,\cdot)\) \(\chi_{6003}(2321,\cdot)\) \(\chi_{6003}(2843,\cdot)\) \(\chi_{6003}(3365,\cdot)\) \(\chi_{6003}(3626,\cdot)\) \(\chi_{6003}(5192,\cdot)\) \(\chi_{6003}(5714,\cdot)\)
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
\((668,3133,4555)\) → \((-1,e\left(\frac{15}{22}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 6003 }(755, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) |
sage: chi.jacobi_sum(n)