from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(322, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([55,3]))
pari: [g,chi] = znchar(Mod(5,322))
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 \(\chi_{161}(5,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 322.o
\(\chi_{322}(5,\cdot)\) \(\chi_{322}(17,\cdot)\) \(\chi_{322}(19,\cdot)\) \(\chi_{322}(33,\cdot)\) \(\chi_{322}(61,\cdot)\) \(\chi_{322}(89,\cdot)\) \(\chi_{322}(103,\cdot)\) \(\chi_{322}(129,\cdot)\) \(\chi_{322}(143,\cdot)\) \(\chi_{322}(145,\cdot)\) \(\chi_{322}(157,\cdot)\) \(\chi_{322}(159,\cdot)\) \(\chi_{322}(171,\cdot)\) \(\chi_{322}(199,\cdot)\) \(\chi_{322}(201,\cdot)\) \(\chi_{322}(227,\cdot)\) \(\chi_{322}(241,\cdot)\) \(\chi_{322}(283,\cdot)\) \(\chi_{322}(297,\cdot)\) \(\chi_{322}(313,\cdot)\)
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
\((185,281)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{1}{22}\right))\)
Values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(25\) | \(27\) |
| \( \chi_{ 322 }(5, a) \) | \(1\) | \(1\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{15}{22}\right)\) |
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)