from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2320, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([0,21,7,17]))
pari: [g,chi] = znchar(Mod(717,2320))
Basic properties
| Modulus: | \(2320\) | |
| Conductor: | \(2320\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2320.et
\(\chi_{2320}(77,\cdot)\) \(\chi_{2320}(213,\cdot)\) \(\chi_{2320}(453,\cdot)\) \(\chi_{2320}(693,\cdot)\) \(\chi_{2320}(717,\cdot)\) \(\chi_{2320}(797,\cdot)\) \(\chi_{2320}(1013,\cdot)\) \(\chi_{2320}(1597,\cdot)\) \(\chi_{2320}(1813,\cdot)\) \(\chi_{2320}(1893,\cdot)\) \(\chi_{2320}(1917,\cdot)\) \(\chi_{2320}(2157,\cdot)\)
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
\((2031,581,1857,321)\) → \((1,-i,i,e\left(\frac{17}{28}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 2320 }(717, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(1\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{3}{28}\right)\) |
sage: chi.jacobi_sum(n)