from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(432, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,9,10]))
pari: [g,chi] = znchar(Mod(5,432))
Basic properties
Modulus: | \(432\) | |
Conductor: | \(432\) | 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 432.bi
\(\chi_{432}(5,\cdot)\) \(\chi_{432}(29,\cdot)\) \(\chi_{432}(77,\cdot)\) \(\chi_{432}(101,\cdot)\) \(\chi_{432}(149,\cdot)\) \(\chi_{432}(173,\cdot)\) \(\chi_{432}(221,\cdot)\) \(\chi_{432}(245,\cdot)\) \(\chi_{432}(293,\cdot)\) \(\chi_{432}(317,\cdot)\) \(\chi_{432}(365,\cdot)\) \(\chi_{432}(389,\cdot)\)
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: | 36.0.5532004127928253705369187176396364210546696053048780432717505515499814912.1 |
Values on generators
\((271,325,353)\) → \((1,i,e\left(\frac{5}{18}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
\( \chi_{ 432 }(5, a) \) | \(-1\) | \(1\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{5}{9}\right)\) |
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)