from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4508, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([0,11,1]))
pari: [g,chi] = znchar(Mod(97,4508))
Basic properties
| Modulus: | \(4508\) | |
| Conductor: | \(161\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{161}(97,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4508.be
\(\chi_{4508}(97,\cdot)\) \(\chi_{4508}(293,\cdot)\) \(\chi_{4508}(881,\cdot)\) \(\chi_{4508}(1077,\cdot)\) \(\chi_{4508}(1469,\cdot)\) \(\chi_{4508}(1861,\cdot)\) \(\chi_{4508}(2057,\cdot)\) \(\chi_{4508}(2449,\cdot)\) \(\chi_{4508}(3625,\cdot)\) \(\chi_{4508}(4017,\cdot)\)
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: | 22.22.78048218870425324004237696277333187889.1 |
Values on generators
\((2255,1473,1569)\) → \((1,-1,e\left(\frac{1}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(25\) | \(27\) |
| \( \chi_{ 4508 }(97, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{15}{22}\right)\) |
sage: chi.jacobi_sum(n)