from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6480, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,27,22,18]))
pari: [g,chi] = znchar(Mod(179,6480))
Basic properties
| Modulus: | \(6480\) | |
| Conductor: | \(2160\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2160}(1859,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6480.dv
\(\chi_{6480}(179,\cdot)\) \(\chi_{6480}(899,\cdot)\) \(\chi_{6480}(1259,\cdot)\) \(\chi_{6480}(1979,\cdot)\) \(\chi_{6480}(2339,\cdot)\) \(\chi_{6480}(3059,\cdot)\) \(\chi_{6480}(3419,\cdot)\) \(\chi_{6480}(4139,\cdot)\) \(\chi_{6480}(4499,\cdot)\) \(\chi_{6480}(5219,\cdot)\) \(\chi_{6480}(5579,\cdot)\) \(\chi_{6480}(6299,\cdot)\)
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: | 36.36.21102921020234122106053112702928025095164093219943162661428472578048000000000000000000.1 |
Values on generators
\((2431,1621,6401,1297)\) → \((-1,-i,e\left(\frac{11}{18}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 6480 }(179, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{8}{9}\right)\) |
sage: chi.jacobi_sum(n)