from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(360000, base_ring=CyclotomicField(6000))
M = H._module
chi = DirichletCharacter(H, M([3000,2625,1000,2916]))
chi.galois_orbit()
[g,chi] = znchar(Mod(83,360000))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(360000\) | |
| Conductor: | \(360000\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(6000\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{6000})$ |
| Fixed field: | Number field defined by a degree 6000 polynomial (not computed) |
First 8 of 1600 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{360000}(83,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{151}{600}\right)\) | \(e\left(\frac{1141}{6000}\right)\) | \(e\left(\frac{2699}{6000}\right)\) | \(e\left(\frac{207}{250}\right)\) | \(e\left(\frac{1421}{2000}\right)\) | \(e\left(\frac{2773}{3000}\right)\) | \(e\left(\frac{4267}{6000}\right)\) | \(e\left(\frac{248}{375}\right)\) | \(e\left(\frac{1663}{2000}\right)\) | \(e\left(\frac{1027}{3000}\right)\) |
| \(\chi_{360000}(347,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{209}{600}\right)\) | \(e\left(\frac{2219}{6000}\right)\) | \(e\left(\frac{2341}{6000}\right)\) | \(e\left(\frac{13}{250}\right)\) | \(e\left(\frac{739}{2000}\right)\) | \(e\left(\frac{1307}{3000}\right)\) | \(e\left(\frac{53}{6000}\right)\) | \(e\left(\frac{82}{375}\right)\) | \(e\left(\frac{817}{2000}\right)\) | \(e\left(\frac{893}{3000}\right)\) |
| \(\chi_{360000}(563,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{143}{600}\right)\) | \(e\left(\frac{4013}{6000}\right)\) | \(e\left(\frac{5107}{6000}\right)\) | \(e\left(\frac{51}{250}\right)\) | \(e\left(\frac{53}{2000}\right)\) | \(e\left(\frac{1589}{3000}\right)\) | \(e\left(\frac{131}{6000}\right)\) | \(e\left(\frac{139}{375}\right)\) | \(e\left(\frac{359}{2000}\right)\) | \(e\left(\frac{1811}{3000}\right)\) |
| \(\chi_{360000}(587,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{133}{600}\right)\) | \(e\left(\frac{703}{6000}\right)\) | \(e\left(\frac{17}{6000}\right)\) | \(e\left(\frac{181}{250}\right)\) | \(e\left(\frac{1943}{2000}\right)\) | \(e\left(\frac{1159}{3000}\right)\) | \(e\left(\frac{2161}{6000}\right)\) | \(e\left(\frac{209}{375}\right)\) | \(e\left(\frac{29}{2000}\right)\) | \(e\left(\frac{241}{3000}\right)\) |
| \(\chi_{360000}(803,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{600}\right)\) | \(e\left(\frac{1729}{6000}\right)\) | \(e\left(\frac{5231}{6000}\right)\) | \(e\left(\frac{33}{250}\right)\) | \(e\left(\frac{1049}{2000}\right)\) | \(e\left(\frac{337}{3000}\right)\) | \(e\left(\frac{1423}{6000}\right)\) | \(e\left(\frac{362}{375}\right)\) | \(e\left(\frac{1747}{2000}\right)\) | \(e\left(\frac{2863}{3000}\right)\) |
| \(\chi_{360000}(1067,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{341}{600}\right)\) | \(e\left(\frac{2231}{6000}\right)\) | \(e\left(\frac{5209}{6000}\right)\) | \(e\left(\frac{137}{250}\right)\) | \(e\left(\frac{1711}{2000}\right)\) | \(e\left(\frac{2543}{3000}\right)\) | \(e\left(\frac{1097}{6000}\right)\) | \(e\left(\frac{268}{375}\right)\) | \(e\left(\frac{533}{2000}\right)\) | \(e\left(\frac{257}{3000}\right)\) |
| \(\chi_{360000}(1283,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{131}{600}\right)\) | \(e\left(\frac{1721}{6000}\right)\) | \(e\left(\frac{3319}{6000}\right)\) | \(e\left(\frac{117}{250}\right)\) | \(e\left(\frac{401}{2000}\right)\) | \(e\left(\frac{2513}{3000}\right)\) | \(e\left(\frac{4727}{6000}\right)\) | \(e\left(\frac{238}{375}\right)\) | \(e\left(\frac{603}{2000}\right)\) | \(e\left(\frac{287}{3000}\right)\) |
| \(\chi_{360000}(1523,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{367}{600}\right)\) | \(e\left(\frac{1597}{6000}\right)\) | \(e\left(\frac{3683}{6000}\right)\) | \(e\left(\frac{169}{250}\right)\) | \(e\left(\frac{357}{2000}\right)\) | \(e\left(\frac{1741}{3000}\right)\) | \(e\left(\frac{1939}{6000}\right)\) | \(e\left(\frac{191}{375}\right)\) | \(e\left(\frac{871}{2000}\right)\) | \(e\left(\frac{859}{3000}\right)\) |