from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4176, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([0,14,14,1]))
pari: [g,chi] = znchar(Mod(89,4176))
Basic properties
| Modulus: | \(4176\) | |
| Conductor: | \(696\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{696}(437,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4176.en
\(\chi_{4176}(89,\cdot)\) \(\chi_{4176}(665,\cdot)\) \(\chi_{4176}(809,\cdot)\) \(\chi_{4176}(1529,\cdot)\) \(\chi_{4176}(1817,\cdot)\) \(\chi_{4176}(1961,\cdot)\) \(\chi_{4176}(2393,\cdot)\) \(\chi_{4176}(2537,\cdot)\) \(\chi_{4176}(2969,\cdot)\) \(\chi_{4176}(3113,\cdot)\) \(\chi_{4176}(3401,\cdot)\) \(\chi_{4176}(4121,\cdot)\)
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: | \(\Q(\zeta_{28})\) |
| Fixed field: | 28.28.64224883807413352704377353768048295264353596334281414672384.1 |
Values on generators
\((1567,1045,929,4033)\) → \((1,-1,-1,e\left(\frac{1}{28}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(31\) | \(35\) |
| \( \chi_{ 4176 }(89, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(i\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) |
sage: chi.jacobi_sum(n)