from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4508, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,44,12]))
pari: [g,chi] = znchar(Mod(165,4508))
Basic properties
| Modulus: | \(4508\) | |
| 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 \(\chi_{161}(4,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4508.bh
\(\chi_{4508}(165,\cdot)\) \(\chi_{4508}(177,\cdot)\) \(\chi_{4508}(361,\cdot)\) \(\chi_{4508}(765,\cdot)\) \(\chi_{4508}(949,\cdot)\) \(\chi_{4508}(961,\cdot)\) \(\chi_{4508}(1145,\cdot)\) \(\chi_{4508}(1549,\cdot)\) \(\chi_{4508}(1733,\cdot)\) \(\chi_{4508}(1941,\cdot)\) \(\chi_{4508}(2125,\cdot)\) \(\chi_{4508}(3117,\cdot)\) \(\chi_{4508}(3301,\cdot)\) \(\chi_{4508}(3509,\cdot)\) \(\chi_{4508}(3693,\cdot)\) \(\chi_{4508}(3705,\cdot)\) \(\chi_{4508}(3889,\cdot)\) \(\chi_{4508}(4097,\cdot)\) \(\chi_{4508}(4281,\cdot)\) \(\chi_{4508}(4489,\cdot)\)
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
\((2255,1473,1569)\) → \((1,e\left(\frac{2}{3}\right),e\left(\frac{2}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(25\) | \(27\) |
| \( \chi_{ 4508 }(165, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{8}{11}\right)\) |
sage: chi.jacobi_sum(n)