from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(322, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,17]))
pari: [g,chi] = znchar(Mod(153,322))
Basic properties
Modulus: | \(322\) | |
Conductor: | \(161\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{161}(153,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 322.k
\(\chi_{322}(83,\cdot)\) \(\chi_{322}(97,\cdot)\) \(\chi_{322}(111,\cdot)\) \(\chi_{322}(125,\cdot)\) \(\chi_{322}(153,\cdot)\) \(\chi_{322}(181,\cdot)\) \(\chi_{322}(195,\cdot)\) \(\chi_{322}(237,\cdot)\) \(\chi_{322}(251,\cdot)\) \(\chi_{322}(293,\cdot)\)
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: | \(\Q(\zeta_{11})\) |
Fixed field: | 22.22.78048218870425324004237696277333187889.1 |
Values on generators
\((185,281)\) → \((-1,e\left(\frac{17}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(25\) | \(27\) |
\( \chi_{ 322 }(153, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) |
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)