from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(855, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([24,27,32]))
pari: [g,chi] = znchar(Mod(43,855))
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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 855.di
\(\chi_{855}(43,\cdot)\) \(\chi_{855}(187,\cdot)\) \(\chi_{855}(358,\cdot)\) \(\chi_{855}(427,\cdot)\) \(\chi_{855}(517,\cdot)\) \(\chi_{855}(598,\cdot)\) \(\chi_{855}(652,\cdot)\) \(\chi_{855}(682,\cdot)\) \(\chi_{855}(688,\cdot)\) \(\chi_{855}(727,\cdot)\) \(\chi_{855}(823,\cdot)\) \(\chi_{855}(853,\cdot)\)
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: | 36.0.49445363514865619772064285277005825358554886701457421415344288356602191925048828125.2 |
Values on generators
\((191,172,496)\) → \((e\left(\frac{2}{3}\right),-i,e\left(\frac{8}{9}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(22\) |
\( \chi_{ 855 }(43, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(-i\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{23}{36}\right)\) |
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)