from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6480, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,9,2,0]))
pari: [g,chi] = znchar(Mod(251,6480))
Basic properties
Modulus: | \(6480\) | |
Conductor: | \(432\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{432}(299,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6480.eq
\(\chi_{6480}(251,\cdot)\) \(\chi_{6480}(611,\cdot)\) \(\chi_{6480}(1331,\cdot)\) \(\chi_{6480}(1691,\cdot)\) \(\chi_{6480}(2411,\cdot)\) \(\chi_{6480}(2771,\cdot)\) \(\chi_{6480}(3491,\cdot)\) \(\chi_{6480}(3851,\cdot)\) \(\chi_{6480}(4571,\cdot)\) \(\chi_{6480}(4931,\cdot)\) \(\chi_{6480}(5651,\cdot)\) \(\chi_{6480}(6011,\cdot)\)
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: | 36.36.5532004127928253705369187176396364210546696053048780432717505515499814912.1 |
Values on generators
\((2431,1621,6401,1297)\) → \((-1,i,e\left(\frac{1}{18}\right),1)\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 6480 }(251, a) \) | \(1\) | \(1\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{4}{9}\right)\) |
sage: chi.jacobi_sum(n)