from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7488, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([0,21,0,20]))
pari: [g,chi] = znchar(Mod(361,7488))
Basic properties
| Modulus: | \(7488\) | |
| Conductor: | \(416\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(24\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{416}(205,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7488.ke
\(\chi_{7488}(361,\cdot)\) \(\chi_{7488}(1369,\cdot)\) \(\chi_{7488}(2233,\cdot)\) \(\chi_{7488}(3241,\cdot)\) \(\chi_{7488}(4105,\cdot)\) \(\chi_{7488}(5113,\cdot)\) \(\chi_{7488}(5977,\cdot)\) \(\chi_{7488}(6985,\cdot)\)
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: | \(\Q(\zeta_{24})\) |
| Fixed field: | 24.24.188216044816745326913150945765287080795084676923392.1 |
Values on generators
\((703,6085,5825,5761)\) → \((1,e\left(\frac{7}{8}\right),1,e\left(\frac{5}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 7488 }(361, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(-i\) | \(e\left(\frac{23}{24}\right)\) | \(-1\) | \(e\left(\frac{7}{24}\right)\) |
sage: chi.jacobi_sum(n)