from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(855, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([12,27,14]))
pari: [g,chi] = znchar(Mod(508,855))
Basic properties
| Modulus: | \(855\) | |
| Conductor: | \(855\) | 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 855.dq
\(\chi_{855}(22,\cdot)\) \(\chi_{855}(193,\cdot)\) \(\chi_{855}(337,\cdot)\) \(\chi_{855}(382,\cdot)\) \(\chi_{855}(412,\cdot)\) \(\chi_{855}(508,\cdot)\) \(\chi_{855}(547,\cdot)\) \(\chi_{855}(553,\cdot)\) \(\chi_{855}(583,\cdot)\) \(\chi_{855}(637,\cdot)\) \(\chi_{855}(718,\cdot)\) \(\chi_{855}(808,\cdot)\)
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: | 36.36.17849776228866488737715206984999102954438314099226129130939288096733391284942626953125.1 |
Values on generators
\((191,172,496)\) → \((e\left(\frac{1}{3}\right),-i,e\left(\frac{7}{18}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(22\) |
| \( \chi_{ 855 }(508, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(1\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)