from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4998, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([0,8,13]))
pari: [g,chi] = znchar(Mod(97,4998))
Basic properties
| Modulus: | \(4998\) | |
| Conductor: | \(119\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{119}(97,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4998.bm
\(\chi_{4998}(97,\cdot)\) \(\chi_{4998}(685,\cdot)\) \(\chi_{4998}(979,\cdot)\) \(\chi_{4998}(1567,\cdot)\) \(\chi_{4998}(2743,\cdot)\) \(\chi_{4998}(3037,\cdot)\) \(\chi_{4998}(4213,\cdot)\) \(\chi_{4998}(4801,\cdot)\)
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: | \(\Q(\zeta_{16})\) |
| Fixed field: | 16.16.16501299269766837593302193.1 |
Values on generators
\((1667,2551,4117)\) → \((1,-1,e\left(\frac{13}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 4998 }(97, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(-i\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) |
sage: chi.jacobi_sum(n)