from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(945, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([14,9,6]))
pari: [g,chi] = znchar(Mod(857,945))
Basic properties
| Modulus: | \(945\) | |
| Conductor: | \(945\) | 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 945.dq
\(\chi_{945}(38,\cdot)\) \(\chi_{945}(68,\cdot)\) \(\chi_{945}(227,\cdot)\) \(\chi_{945}(257,\cdot)\) \(\chi_{945}(353,\cdot)\) \(\chi_{945}(383,\cdot)\) \(\chi_{945}(542,\cdot)\) \(\chi_{945}(572,\cdot)\) \(\chi_{945}(668,\cdot)\) \(\chi_{945}(698,\cdot)\) \(\chi_{945}(857,\cdot)\) \(\chi_{945}(887,\cdot)\)
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: | 36.0.1465697146251004474650413649141163426679210854529430772252466557978101074695587158203125.1 |
Values on generators
\((596,757,136)\) → \((e\left(\frac{7}{18}\right),i,e\left(\frac{1}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
| \( \chi_{ 945 }(857, a) \) | \(-1\) | \(1\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(i\) | \(1\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{13}{36}\right)\) |
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)