from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(855, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([12,9,20]))
pari: [g,chi] = znchar(Mod(112,855))
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.dr
\(\chi_{855}(112,\cdot)\) \(\chi_{855}(142,\cdot)\) \(\chi_{855}(157,\cdot)\) \(\chi_{855}(232,\cdot)\) \(\chi_{855}(283,\cdot)\) \(\chi_{855}(313,\cdot)\) \(\chi_{855}(328,\cdot)\) \(\chi_{855}(367,\cdot)\) \(\chi_{855}(403,\cdot)\) \(\chi_{855}(472,\cdot)\) \(\chi_{855}(538,\cdot)\) \(\chi_{855}(643,\cdot)\)
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: | 36.0.49445363514865619772064285277005825358554886701457421415344288356602191925048828125.1 |
Values on generators
\((191,172,496)\) → \((e\left(\frac{1}{3}\right),i,e\left(\frac{5}{9}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(22\) |
\( \chi_{ 855 }(112, a) \) | \(-1\) | \(1\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(1\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{5}{36}\right)\) |
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)