from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6384, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,27,18,12,10]))
pari: [g,chi] = znchar(Mod(317,6384))
Basic properties
| Modulus: | \(6384\) | |
| Conductor: | \(6384\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6384.ne
\(\chi_{6384}(317,\cdot)\) \(\chi_{6384}(485,\cdot)\) \(\chi_{6384}(725,\cdot)\) \(\chi_{6384}(2237,\cdot)\) \(\chi_{6384}(2333,\cdot)\) \(\chi_{6384}(2909,\cdot)\) \(\chi_{6384}(3509,\cdot)\) \(\chi_{6384}(3677,\cdot)\) \(\chi_{6384}(3917,\cdot)\) \(\chi_{6384}(5429,\cdot)\) \(\chi_{6384}(5525,\cdot)\) \(\chi_{6384}(6101,\cdot)\)
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
\((799,4789,2129,913,1009)\) → \((1,-i,-1,e\left(\frac{1}{3}\right),e\left(\frac{5}{18}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 6384 }(317, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(-1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{11}{18}\right)\) |
sage: chi.jacobi_sum(n)