from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6480, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,18,34,27]))
pari: [g,chi] = znchar(Mod(233,6480))
Basic properties
| Modulus: | \(6480\) | |
| Conductor: | \(1080\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1080}(1013,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6480.dz
\(\chi_{6480}(233,\cdot)\) \(\chi_{6480}(953,\cdot)\) \(\chi_{6480}(1097,\cdot)\) \(\chi_{6480}(1817,\cdot)\) \(\chi_{6480}(2393,\cdot)\) \(\chi_{6480}(3113,\cdot)\) \(\chi_{6480}(3257,\cdot)\) \(\chi_{6480}(3977,\cdot)\) \(\chi_{6480}(4553,\cdot)\) \(\chi_{6480}(5273,\cdot)\) \(\chi_{6480}(5417,\cdot)\) \(\chi_{6480}(6137,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
| Fixed field: | 36.36.1171447440175506066520511744440292972455846710607872000000000000000000000000000.1 |
Values on generators
\((2431,1621,6401,1297)\) → \((1,-1,e\left(\frac{17}{18}\right),-i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 6480 }(233, a) \) | \(1\) | \(1\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{18}\right)\) |
sage: chi.jacobi_sum(n)