from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(786, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([13,21]))
pari: [g,chi] = znchar(Mod(47,786))
Basic properties
Modulus: | \(786\) | |
Conductor: | \(393\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(26\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{393}(47,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 786.k
\(\chi_{786}(47,\cdot)\) \(\chi_{786}(71,\cdot)\) \(\chi_{786}(149,\cdot)\) \(\chi_{786}(155,\cdot)\) \(\chi_{786}(281,\cdot)\) \(\chi_{786}(341,\cdot)\) \(\chi_{786}(425,\cdot)\) \(\chi_{786}(461,\cdot)\) \(\chi_{786}(479,\cdot)\) \(\chi_{786}(485,\cdot)\) \(\chi_{786}(575,\cdot)\) \(\chi_{786}(593,\cdot)\)
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: | \(\Q(\zeta_{13})\) |
Fixed field: | 26.26.136256962349903875302212256879702783874125319329547338554073.1 |
Values on generators
\((263,133)\) → \((-1,e\left(\frac{21}{26}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
\( \chi_{ 786 }(47, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{11}{26}\right)\) |
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)