from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1925, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([27,40,30]))
pari: [g,chi] = znchar(Mod(1187,1925))
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.fy
\(\chi_{1925}(142,\cdot)\) \(\chi_{1925}(263,\cdot)\) \(\chi_{1925}(373,\cdot)\) \(\chi_{1925}(417,\cdot)\) \(\chi_{1925}(527,\cdot)\) \(\chi_{1925}(648,\cdot)\) \(\chi_{1925}(758,\cdot)\) \(\chi_{1925}(802,\cdot)\) \(\chi_{1925}(912,\cdot)\) \(\chi_{1925}(1033,\cdot)\) \(\chi_{1925}(1187,\cdot)\) \(\chi_{1925}(1297,\cdot)\) \(\chi_{1925}(1528,\cdot)\) \(\chi_{1925}(1572,\cdot)\) \(\chi_{1925}(1803,\cdot)\) \(\chi_{1925}(1913,\cdot)\)
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
\((1002,276,1751)\) → \((e\left(\frac{9}{20}\right),e\left(\frac{2}{3}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(12\) | \(13\) | \(16\) | \(17\) |
| \( \chi_{ 1925 }(1187, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{1}{60}\right)\) |
sage: chi.jacobi_sum(n)