from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(864, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,9,34]))
pari: [g,chi] = znchar(Mod(743,864))
Basic properties
Modulus: | \(864\) | |
Conductor: | \(432\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{432}(203,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 864.br
\(\chi_{864}(23,\cdot)\) \(\chi_{864}(119,\cdot)\) \(\chi_{864}(167,\cdot)\) \(\chi_{864}(263,\cdot)\) \(\chi_{864}(311,\cdot)\) \(\chi_{864}(407,\cdot)\) \(\chi_{864}(455,\cdot)\) \(\chi_{864}(551,\cdot)\) \(\chi_{864}(599,\cdot)\) \(\chi_{864}(695,\cdot)\) \(\chi_{864}(743,\cdot)\) \(\chi_{864}(839,\cdot)\)
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: | 36.36.5532004127928253705369187176396364210546696053048780432717505515499814912.1 |
Values on generators
\((703,325,353)\) → \((-1,i,e\left(\frac{17}{18}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
\( \chi_{ 864 }(743, a) \) | \(1\) | \(1\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) |
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)