from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1337, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([19,23]))
pari: [g,chi] = znchar(Mod(734,1337))
Basic properties
| Modulus: | \(1337\) | |
| Conductor: | \(1337\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(38\) | 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 1337.t
\(\chi_{1337}(41,\cdot)\) \(\chi_{1337}(55,\cdot)\) \(\chi_{1337}(139,\cdot)\) \(\chi_{1337}(202,\cdot)\) \(\chi_{1337}(377,\cdot)\) \(\chi_{1337}(419,\cdot)\) \(\chi_{1337}(587,\cdot)\) \(\chi_{1337}(643,\cdot)\) \(\chi_{1337}(657,\cdot)\) \(\chi_{1337}(734,\cdot)\) \(\chi_{1337}(923,\cdot)\) \(\chi_{1337}(930,\cdot)\) \(\chi_{1337}(986,\cdot)\) \(\chi_{1337}(993,\cdot)\) \(\chi_{1337}(1021,\cdot)\) \(\chi_{1337}(1077,\cdot)\) \(\chi_{1337}(1140,\cdot)\) \(\chi_{1337}(1301,\cdot)\)
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: | \(\Q(\zeta_{19})\) |
| Fixed field: | Number field defined by a degree 38 polynomial |
Values on generators
\((192,974)\) → \((-1,e\left(\frac{23}{38}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 1337 }(734, a) \) | \(1\) | \(1\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{13}{38}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{17}{38}\right)\) | \(e\left(\frac{37}{38}\right)\) |
sage: chi.jacobi_sum(n)