from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4176, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([0,21,14,3]))
pari: [g,chi] = znchar(Mod(269,4176))
Basic properties
| Modulus: | \(4176\) | |
| Conductor: | \(1392\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1392}(269,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4176.em
\(\chi_{4176}(269,\cdot)\) \(\chi_{4176}(485,\cdot)\) \(\chi_{4176}(773,\cdot)\) \(\chi_{4176}(1493,\cdot)\) \(\chi_{4176}(1709,\cdot)\) \(\chi_{4176}(1853,\cdot)\) \(\chi_{4176}(1925,\cdot)\) \(\chi_{4176}(3005,\cdot)\) \(\chi_{4176}(3077,\cdot)\) \(\chi_{4176}(3221,\cdot)\) \(\chi_{4176}(3437,\cdot)\) \(\chi_{4176}(4157,\cdot)\)
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: | Number field defined by a degree 28 polynomial |
Values on generators
\((1567,1045,929,4033)\) → \((1,-i,-1,e\left(\frac{3}{28}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(31\) | \(35\) |
| \( \chi_{ 4176 }(269, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(-i\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{11}{28}\right)\) |
sage: chi.jacobi_sum(n)