from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1375, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([38,40]))
pari: [g,chi] = znchar(Mod(1136,1375))
Basic properties
| Modulus: | \(1375\) | |
| Conductor: | \(1375\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(25\) | 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 1375.bt
\(\chi_{1375}(36,\cdot)\) \(\chi_{1375}(181,\cdot)\) \(\chi_{1375}(191,\cdot)\) \(\chi_{1375}(196,\cdot)\) \(\chi_{1375}(311,\cdot)\) \(\chi_{1375}(456,\cdot)\) \(\chi_{1375}(466,\cdot)\) \(\chi_{1375}(471,\cdot)\) \(\chi_{1375}(586,\cdot)\) \(\chi_{1375}(731,\cdot)\) \(\chi_{1375}(741,\cdot)\) \(\chi_{1375}(746,\cdot)\) \(\chi_{1375}(861,\cdot)\) \(\chi_{1375}(1006,\cdot)\) \(\chi_{1375}(1016,\cdot)\) \(\chi_{1375}(1021,\cdot)\) \(\chi_{1375}(1136,\cdot)\) \(\chi_{1375}(1281,\cdot)\) \(\chi_{1375}(1291,\cdot)\) \(\chi_{1375}(1296,\cdot)\)
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: | \(\Q(\zeta_{25})\) |
| Fixed field: | 25.25.227936564389216769066972959924266550757465665810741484165191650390625.3 |
Values on generators
\((1002,376)\) → \((e\left(\frac{19}{25}\right),e\left(\frac{4}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 1375 }(1136, a) \) | \(1\) | \(1\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{19}{25}\right)\) |
sage: chi.jacobi_sum(n)