from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6900, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,0,11,17]))
pari: [g,chi] = znchar(Mod(199,6900))
Basic properties
| Modulus: | \(6900\) | |
| Conductor: | \(460\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{460}(199,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6900.cg
\(\chi_{6900}(199,\cdot)\) \(\chi_{6900}(799,\cdot)\) \(\chi_{6900}(1399,\cdot)\) \(\chi_{6900}(1699,\cdot)\) \(\chi_{6900}(1999,\cdot)\) \(\chi_{6900}(4099,\cdot)\) \(\chi_{6900}(4699,\cdot)\) \(\chi_{6900}(5599,\cdot)\) \(\chi_{6900}(5899,\cdot)\) \(\chi_{6900}(6799,\cdot)\)
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: | 22.22.8083780427918435509708715954790400000000000.1 |
Values on generators
\((3451,4601,277,1201)\) → \((-1,1,-1,e\left(\frac{17}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 6900 }(199, a) \) | \(1\) | \(1\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{19}{22}\right)\) |
sage: chi.jacobi_sum(n)