from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6384, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,9,0,0,16]))
pari: [g,chi] = znchar(Mod(85,6384))
Basic properties
| Modulus: | \(6384\) | |
| Conductor: | \(304\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{304}(85,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6384.nk
\(\chi_{6384}(85,\cdot)\) \(\chi_{6384}(253,\cdot)\) \(\chi_{6384}(757,\cdot)\) \(\chi_{6384}(1429,\cdot)\) \(\chi_{6384}(1765,\cdot)\) \(\chi_{6384}(2437,\cdot)\) \(\chi_{6384}(3277,\cdot)\) \(\chi_{6384}(3445,\cdot)\) \(\chi_{6384}(3949,\cdot)\) \(\chi_{6384}(4621,\cdot)\) \(\chi_{6384}(4957,\cdot)\) \(\chi_{6384}(5629,\cdot)\)
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: | 36.36.52733281945045886724167383478270850720626086921526306402773390818541568.1 |
Values on generators
\((799,4789,2129,913,1009)\) → \((1,i,1,1,e\left(\frac{4}{9}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 6384 }(85, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(i\) | \(e\left(\frac{5}{18}\right)\) |
sage: chi.jacobi_sum(n)