from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1375, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([29,15]))
pari: [g,chi] = znchar(Mod(1119,1375))
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.by
\(\chi_{1375}(19,\cdot)\) \(\chi_{1375}(29,\cdot)\) \(\chi_{1375}(189,\cdot)\) \(\chi_{1375}(259,\cdot)\) \(\chi_{1375}(294,\cdot)\) \(\chi_{1375}(304,\cdot)\) \(\chi_{1375}(464,\cdot)\) \(\chi_{1375}(534,\cdot)\) \(\chi_{1375}(569,\cdot)\) \(\chi_{1375}(579,\cdot)\) \(\chi_{1375}(739,\cdot)\) \(\chi_{1375}(809,\cdot)\) \(\chi_{1375}(844,\cdot)\) \(\chi_{1375}(854,\cdot)\) \(\chi_{1375}(1014,\cdot)\) \(\chi_{1375}(1084,\cdot)\) \(\chi_{1375}(1119,\cdot)\) \(\chi_{1375}(1129,\cdot)\) \(\chi_{1375}(1289,\cdot)\) \(\chi_{1375}(1359,\cdot)\)
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
\((1002,376)\) → \((e\left(\frac{29}{50}\right),e\left(\frac{3}{10}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
\( \chi_{ 1375 }(1119, a) \) | \(-1\) | \(1\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{17}{50}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{7}{25}\right)\) |
sage: chi.jacobi_sum(n)