from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(322, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([44,27]))
pari: [g,chi] = znchar(Mod(11,322))
Basic properties
| Modulus: | \(322\) | |
| Conductor: | \(161\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{161}(11,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 322.p
\(\chi_{322}(11,\cdot)\) \(\chi_{322}(37,\cdot)\) \(\chi_{322}(51,\cdot)\) \(\chi_{322}(53,\cdot)\) \(\chi_{322}(65,\cdot)\) \(\chi_{322}(67,\cdot)\) \(\chi_{322}(79,\cdot)\) \(\chi_{322}(107,\cdot)\) \(\chi_{322}(109,\cdot)\) \(\chi_{322}(135,\cdot)\) \(\chi_{322}(149,\cdot)\) \(\chi_{322}(191,\cdot)\) \(\chi_{322}(205,\cdot)\) \(\chi_{322}(221,\cdot)\) \(\chi_{322}(235,\cdot)\) \(\chi_{322}(247,\cdot)\) \(\chi_{322}(249,\cdot)\) \(\chi_{322}(263,\cdot)\) \(\chi_{322}(291,\cdot)\) \(\chi_{322}(319,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
| Fixed field: | Number field defined by a degree 66 polynomial |
Values on generators
\((185,281)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{9}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(25\) | \(27\) |
| \( \chi_{ 322 }(11, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{7}{11}\right)\) |
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)