from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1375, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([8,5]))
pari: [g,chi] = znchar(Mod(1036,1375))
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.cg
\(\chi_{1375}(116,\cdot)\) \(\chi_{1375}(206,\cdot)\) \(\chi_{1375}(211,\cdot)\) \(\chi_{1375}(271,\cdot)\) \(\chi_{1375}(391,\cdot)\) \(\chi_{1375}(481,\cdot)\) \(\chi_{1375}(486,\cdot)\) \(\chi_{1375}(546,\cdot)\) \(\chi_{1375}(666,\cdot)\) \(\chi_{1375}(756,\cdot)\) \(\chi_{1375}(761,\cdot)\) \(\chi_{1375}(821,\cdot)\) \(\chi_{1375}(941,\cdot)\) \(\chi_{1375}(1031,\cdot)\) \(\chi_{1375}(1036,\cdot)\) \(\chi_{1375}(1096,\cdot)\) \(\chi_{1375}(1216,\cdot)\) \(\chi_{1375}(1306,\cdot)\) \(\chi_{1375}(1311,\cdot)\) \(\chi_{1375}(1371,\cdot)\)
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
\((1002,376)\) → \((e\left(\frac{4}{25}\right),e\left(\frac{1}{10}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
\( \chi_{ 1375 }(1036, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{50}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{17}{50}\right)\) | \(e\left(\frac{14}{25}\right)\) |
sage: chi.jacobi_sum(n)