from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1375, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([33,0]))
pari: [g,chi] = znchar(Mod(89,1375))
Basic properties
Modulus: | \(1375\) | |
Conductor: | \(125\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(50\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{125}(89,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1375.ce
\(\chi_{1375}(34,\cdot)\) \(\chi_{1375}(89,\cdot)\) \(\chi_{1375}(144,\cdot)\) \(\chi_{1375}(254,\cdot)\) \(\chi_{1375}(309,\cdot)\) \(\chi_{1375}(364,\cdot)\) \(\chi_{1375}(419,\cdot)\) \(\chi_{1375}(529,\cdot)\) \(\chi_{1375}(584,\cdot)\) \(\chi_{1375}(639,\cdot)\) \(\chi_{1375}(694,\cdot)\) \(\chi_{1375}(804,\cdot)\) \(\chi_{1375}(859,\cdot)\) \(\chi_{1375}(914,\cdot)\) \(\chi_{1375}(969,\cdot)\) \(\chi_{1375}(1079,\cdot)\) \(\chi_{1375}(1134,\cdot)\) \(\chi_{1375}(1189,\cdot)\) \(\chi_{1375}(1244,\cdot)\) \(\chi_{1375}(1354,\cdot)\)
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
\((1002,376)\) → \((e\left(\frac{33}{50}\right),1)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
\( \chi_{ 1375 }(89, a) \) | \(1\) | \(1\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{37}{50}\right)\) | \(e\left(\frac{19}{25}\right)\) |
sage: chi.jacobi_sum(n)