from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1375, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([6,25]))
pari: [g,chi] = znchar(Mod(846,1375))
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.ch
\(\chi_{1375}(21,\cdot)\) \(\chi_{1375}(131,\cdot)\) \(\chi_{1375}(186,\cdot)\) \(\chi_{1375}(241,\cdot)\) \(\chi_{1375}(296,\cdot)\) \(\chi_{1375}(406,\cdot)\) \(\chi_{1375}(461,\cdot)\) \(\chi_{1375}(516,\cdot)\) \(\chi_{1375}(571,\cdot)\) \(\chi_{1375}(681,\cdot)\) \(\chi_{1375}(736,\cdot)\) \(\chi_{1375}(791,\cdot)\) \(\chi_{1375}(846,\cdot)\) \(\chi_{1375}(956,\cdot)\) \(\chi_{1375}(1011,\cdot)\) \(\chi_{1375}(1066,\cdot)\) \(\chi_{1375}(1121,\cdot)\) \(\chi_{1375}(1231,\cdot)\) \(\chi_{1375}(1286,\cdot)\) \(\chi_{1375}(1341,\cdot)\)
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
\((1002,376)\) → \((e\left(\frac{3}{25}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
\( \chi_{ 1375 }(846, a) \) | \(-1\) | \(1\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{8}{25}\right)\) |
sage: chi.jacobi_sum(n)