from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5796, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,0,22,42]))
pari: [g,chi] = znchar(Mod(289,5796))
Basic properties
| Modulus: | \(5796\) | |
| Conductor: | \(161\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(33\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{161}(128,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5796.dt
\(\chi_{5796}(289,\cdot)\) \(\chi_{5796}(361,\cdot)\) \(\chi_{5796}(541,\cdot)\) \(\chi_{5796}(1117,\cdot)\) \(\chi_{5796}(1297,\cdot)\) \(\chi_{5796}(1369,\cdot)\) \(\chi_{5796}(1549,\cdot)\) \(\chi_{5796}(2053,\cdot)\) \(\chi_{5796}(2125,\cdot)\) \(\chi_{5796}(2377,\cdot)\) \(\chi_{5796}(2557,\cdot)\) \(\chi_{5796}(2809,\cdot)\) \(\chi_{5796}(2881,\cdot)\) \(\chi_{5796}(3061,\cdot)\) \(\chi_{5796}(3385,\cdot)\) \(\chi_{5796}(3637,\cdot)\) \(\chi_{5796}(3889,\cdot)\) \(\chi_{5796}(4825,\cdot)\) \(\chi_{5796}(5329,\cdot)\) \(\chi_{5796}(5653,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
| Fixed field: | 33.33.277966181338944111003326058293667039541136678070715028736001.1 |
Values on generators
\((2899,1289,829,4789)\) → \((1,1,e\left(\frac{1}{3}\right),e\left(\frac{7}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 5796 }(289, a) \) | \(1\) | \(1\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{7}{11}\right)\) |
sage: chi.jacobi_sum(n)