from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6384, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,27,0,18,10]))
pari: [g,chi] = znchar(Mod(13,6384))
Basic properties
| Modulus: | \(6384\) | |
| Conductor: | \(2128\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2128}(13,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6384.nt
\(\chi_{6384}(13,\cdot)\) \(\chi_{6384}(181,\cdot)\) \(\chi_{6384}(1021,\cdot)\) \(\chi_{6384}(1693,\cdot)\) \(\chi_{6384}(2029,\cdot)\) \(\chi_{6384}(2701,\cdot)\) \(\chi_{6384}(3205,\cdot)\) \(\chi_{6384}(3373,\cdot)\) \(\chi_{6384}(4213,\cdot)\) \(\chi_{6384}(4885,\cdot)\) \(\chi_{6384}(5221,\cdot)\) \(\chi_{6384}(5893,\cdot)\)
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: | Number field defined by a degree 36 polynomial |
Values on generators
\((799,4789,2129,913,1009)\) → \((1,-i,1,-1,e\left(\frac{5}{18}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 6384 }(13, a) \) | \(1\) | \(1\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(i\) | \(e\left(\frac{11}{18}\right)\) |
sage: chi.jacobi_sum(n)