from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8008, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([0,6,2,0,5]))
pari: [g,chi] = znchar(Mod(45,8008))
Basic properties
| Modulus: | \(8008\) | |
| Conductor: | \(728\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(12\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{728}(45,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8008.jd
\(\chi_{8008}(45,\cdot)\) \(\chi_{8008}(4093,\cdot)\) \(\chi_{8008}(4973,\cdot)\) \(\chi_{8008}(7789,\cdot)\)
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: | \(\Q(\zeta_{12})\) |
| Fixed field: | 12.12.132708028528339108551196672.2 |
Values on generators
\((6007,4005,3433,4369,4929)\) → \((1,-1,e\left(\frac{1}{6}\right),1,e\left(\frac{5}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) | \(29\) |
| \( \chi_{ 8008 }(45, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(-1\) | \(e\left(\frac{1}{6}\right)\) | \(1\) | \(e\left(\frac{1}{6}\right)\) |
sage: chi.jacobi_sum(n)