from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8450, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([0,16]))
pari: [g,chi] = znchar(Mod(651,8450))
Basic properties
| Modulus: | \(8450\) | |
| Conductor: | \(169\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(13\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{169}(144,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8450.y
\(\chi_{8450}(651,\cdot)\) \(\chi_{8450}(1301,\cdot)\) \(\chi_{8450}(1951,\cdot)\) \(\chi_{8450}(2601,\cdot)\) \(\chi_{8450}(3251,\cdot)\) \(\chi_{8450}(3901,\cdot)\) \(\chi_{8450}(4551,\cdot)\) \(\chi_{8450}(5201,\cdot)\) \(\chi_{8450}(5851,\cdot)\) \(\chi_{8450}(6501,\cdot)\) \(\chi_{8450}(7151,\cdot)\) \(\chi_{8450}(7801,\cdot)\)
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: | \(\Q(\zeta_{13})\) |
| Fixed field: | 13.13.542800770374370512771595361.1 |
Values on generators
\((677,3551)\) → \((1,e\left(\frac{8}{13}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 8450 }(651, a) \) | \(1\) | \(1\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(1\) | \(e\left(\frac{2}{13}\right)\) | \(1\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) |
sage: chi.jacobi_sum(n)