from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6900, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,11,0,2]))
pari: [g,chi] = znchar(Mod(1451,6900))
Basic properties
| Modulus: | \(6900\) | |
| Conductor: | \(276\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{276}(71,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6900.cf
\(\chi_{6900}(1451,\cdot)\) \(\chi_{6900}(1751,\cdot)\) \(\chi_{6900}(2051,\cdot)\) \(\chi_{6900}(2651,\cdot)\) \(\chi_{6900}(3251,\cdot)\) \(\chi_{6900}(3551,\cdot)\) \(\chi_{6900}(4451,\cdot)\) \(\chi_{6900}(4751,\cdot)\) \(\chi_{6900}(5651,\cdot)\) \(\chi_{6900}(6251,\cdot)\)
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
\((3451,4601,277,1201)\) → \((-1,-1,1,e\left(\frac{1}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 6900 }(1451, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{21}{22}\right)\) |
sage: chi.jacobi_sum(n)