from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6384, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,27,18,0,34]))
pari: [g,chi] = znchar(Mod(29,6384))
Basic properties
| Modulus: | \(6384\) | |
| Conductor: | \(912\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{912}(29,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6384.od
\(\chi_{6384}(29,\cdot)\) \(\chi_{6384}(869,\cdot)\) \(\chi_{6384}(1541,\cdot)\) \(\chi_{6384}(1877,\cdot)\) \(\chi_{6384}(2549,\cdot)\) \(\chi_{6384}(3053,\cdot)\) \(\chi_{6384}(3221,\cdot)\) \(\chi_{6384}(4061,\cdot)\) \(\chi_{6384}(4733,\cdot)\) \(\chi_{6384}(5069,\cdot)\) \(\chi_{6384}(5741,\cdot)\) \(\chi_{6384}(6245,\cdot)\)
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: | 36.36.7375213349858562030923708432825799203687776914247215677306393587785801919440617472.1 |
Values on generators
\((799,4789,2129,913,1009)\) → \((1,-i,-1,1,e\left(\frac{17}{18}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 6384 }(29, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(i\) | \(e\left(\frac{5}{18}\right)\) |
sage: chi.jacobi_sum(n)