from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8050, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,0,12]))
pari: [g,chi] = znchar(Mod(5699,8050))
Basic properties
| Modulus: | \(8050\) | |
| 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 \(\chi_{115}(64,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8050.bp
\(\chi_{8050}(449,\cdot)\) \(\chi_{8050}(1499,\cdot)\) \(\chi_{8050}(1849,\cdot)\) \(\chi_{8050}(3249,\cdot)\) \(\chi_{8050}(3949,\cdot)\) \(\chi_{8050}(4649,\cdot)\) \(\chi_{8050}(4999,\cdot)\) \(\chi_{8050}(5349,\cdot)\) \(\chi_{8050}(5699,\cdot)\) \(\chi_{8050}(7799,\cdot)\)
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
\((2577,1151,3501)\) → \((-1,1,e\left(\frac{6}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(27\) | \(29\) | \(31\) | \(33\) |
| \( \chi_{ 8050 }(5699, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{3}{22}\right)\) |
sage: chi.jacobi_sum(n)