from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1380, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([0,0,11,14]))
pari: [g,chi] = znchar(Mod(289,1380))
Basic properties
| Modulus: | \(1380\) | |
| Conductor: | \(115\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{115}(59,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1380.bj
\(\chi_{1380}(49,\cdot)\) \(\chi_{1380}(169,\cdot)\) \(\chi_{1380}(289,\cdot)\) \(\chi_{1380}(349,\cdot)\) \(\chi_{1380}(409,\cdot)\) \(\chi_{1380}(469,\cdot)\) \(\chi_{1380}(949,\cdot)\) \(\chi_{1380}(1129,\cdot)\) \(\chi_{1380}(1189,\cdot)\) \(\chi_{1380}(1369,\cdot)\)
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: | 22.22.83796671451884098775580820361328125.1 |
Values on generators
\((691,461,277,1201)\) → \((1,1,-1,e\left(\frac{7}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 1380 }(289, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{15}{22}\right)\) |
sage: chi.jacobi_sum(n)