from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1375, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([21,10]))
pari: [g,chi] = znchar(Mod(1104,1375))
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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1375.cf
\(\chi_{1375}(4,\cdot)\) \(\chi_{1375}(64,\cdot)\) \(\chi_{1375}(69,\cdot)\) \(\chi_{1375}(159,\cdot)\) \(\chi_{1375}(279,\cdot)\) \(\chi_{1375}(339,\cdot)\) \(\chi_{1375}(344,\cdot)\) \(\chi_{1375}(434,\cdot)\) \(\chi_{1375}(554,\cdot)\) \(\chi_{1375}(614,\cdot)\) \(\chi_{1375}(619,\cdot)\) \(\chi_{1375}(709,\cdot)\) \(\chi_{1375}(829,\cdot)\) \(\chi_{1375}(889,\cdot)\) \(\chi_{1375}(894,\cdot)\) \(\chi_{1375}(984,\cdot)\) \(\chi_{1375}(1104,\cdot)\) \(\chi_{1375}(1164,\cdot)\) \(\chi_{1375}(1169,\cdot)\) \(\chi_{1375}(1259,\cdot)\)
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
\((1002,376)\) → \((e\left(\frac{21}{50}\right),e\left(\frac{1}{5}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
\( \chi_{ 1375 }(1104, a) \) | \(1\) | \(1\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{18}{25}\right)\) |
sage: chi.jacobi_sum(n)