from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1925, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([33,40,42]))
pari: [g,chi] = znchar(Mod(1173,1925))
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.gl
\(\chi_{1925}(72,\cdot)\) \(\chi_{1925}(233,\cdot)\) \(\chi_{1925}(303,\cdot)\) \(\chi_{1925}(338,\cdot)\) \(\chi_{1925}(347,\cdot)\) \(\chi_{1925}(508,\cdot)\) \(\chi_{1925}(578,\cdot)\) \(\chi_{1925}(613,\cdot)\) \(\chi_{1925}(898,\cdot)\) \(\chi_{1925}(1052,\cdot)\) \(\chi_{1925}(1173,\cdot)\) \(\chi_{1925}(1262,\cdot)\) \(\chi_{1925}(1327,\cdot)\) \(\chi_{1925}(1537,\cdot)\) \(\chi_{1925}(1542,\cdot)\) \(\chi_{1925}(1817,\cdot)\)
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
\((1002,276,1751)\) → \((e\left(\frac{11}{20}\right),e\left(\frac{2}{3}\right),e\left(\frac{7}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(12\) | \(13\) | \(16\) | \(17\) |
| \( \chi_{ 1925 }(1173, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(-i\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{60}\right)\) |
sage: chi.jacobi_sum(n)