from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1925, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([39,10,24]))
pari: [g,chi] = znchar(Mod(192,1925))
Basic properties
| Modulus: | \(1925\) | |
| Conductor: | \(1925\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | 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 1925.ge
\(\chi_{1925}(192,\cdot)\) \(\chi_{1925}(213,\cdot)\) \(\chi_{1925}(467,\cdot)\) \(\chi_{1925}(488,\cdot)\) \(\chi_{1925}(647,\cdot)\) \(\chi_{1925}(773,\cdot)\) \(\chi_{1925}(808,\cdot)\) \(\chi_{1925}(878,\cdot)\) \(\chi_{1925}(922,\cdot)\) \(\chi_{1925}(927,\cdot)\) \(\chi_{1925}(1048,\cdot)\) \(\chi_{1925}(1083,\cdot)\) \(\chi_{1925}(1137,\cdot)\) \(\chi_{1925}(1153,\cdot)\) \(\chi_{1925}(1202,\cdot)\) \(\chi_{1925}(1412,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((1002,276,1751)\) → \((e\left(\frac{13}{20}\right),e\left(\frac{1}{6}\right),e\left(\frac{2}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(12\) | \(13\) | \(16\) | \(17\) |
| \( \chi_{ 1925 }(192, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(i\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{13}{60}\right)\) |
sage: chi.jacobi_sum(n)