from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1375, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([22,15]))
pari: [g,chi] = znchar(Mod(41,1375))
Basic properties
Modulus: | \(1375\) | |
Conductor: | \(1375\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(50\) | 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)
|
Galois orbit 1375.bw
\(\chi_{1375}(41,\cdot)\) \(\chi_{1375}(156,\cdot)\) \(\chi_{1375}(161,\cdot)\) \(\chi_{1375}(171,\cdot)\) \(\chi_{1375}(316,\cdot)\) \(\chi_{1375}(431,\cdot)\) \(\chi_{1375}(436,\cdot)\) \(\chi_{1375}(446,\cdot)\) \(\chi_{1375}(591,\cdot)\) \(\chi_{1375}(706,\cdot)\) \(\chi_{1375}(711,\cdot)\) \(\chi_{1375}(721,\cdot)\) \(\chi_{1375}(866,\cdot)\) \(\chi_{1375}(981,\cdot)\) \(\chi_{1375}(986,\cdot)\) \(\chi_{1375}(996,\cdot)\) \(\chi_{1375}(1141,\cdot)\) \(\chi_{1375}(1256,\cdot)\) \(\chi_{1375}(1261,\cdot)\) \(\chi_{1375}(1271,\cdot)\)
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: | Number field defined by a degree 50 polynomial |
Values on generators
\((1002,376)\) → \((e\left(\frac{11}{25}\right),e\left(\frac{3}{10}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
\( \chi_{ 1375 }(41, a) \) | \(-1\) | \(1\) | \(e\left(\frac{37}{50}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(-1\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{6}{25}\right)\) |
sage: chi.jacobi_sum(n)