from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1375, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([27,25]))
pari: [g,chi] = znchar(Mod(109,1375))
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.bz
\(\chi_{1375}(54,\cdot)\) \(\chi_{1375}(109,\cdot)\) \(\chi_{1375}(164,\cdot)\) \(\chi_{1375}(219,\cdot)\) \(\chi_{1375}(329,\cdot)\) \(\chi_{1375}(384,\cdot)\) \(\chi_{1375}(439,\cdot)\) \(\chi_{1375}(494,\cdot)\) \(\chi_{1375}(604,\cdot)\) \(\chi_{1375}(659,\cdot)\) \(\chi_{1375}(714,\cdot)\) \(\chi_{1375}(769,\cdot)\) \(\chi_{1375}(879,\cdot)\) \(\chi_{1375}(934,\cdot)\) \(\chi_{1375}(989,\cdot)\) \(\chi_{1375}(1044,\cdot)\) \(\chi_{1375}(1154,\cdot)\) \(\chi_{1375}(1209,\cdot)\) \(\chi_{1375}(1264,\cdot)\) \(\chi_{1375}(1319,\cdot)\)
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
\((1002,376)\) → \((e\left(\frac{27}{50}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
\( \chi_{ 1375 }(109, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{11}{25}\right)\) |
sage: chi.jacobi_sum(n)