from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(882, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,25]))
pari: [g,chi] = znchar(Mod(17,882))
Basic properties
Modulus: | \(882\) | |
Conductor: | \(147\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{147}(17,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 882.bl
\(\chi_{882}(17,\cdot)\) \(\chi_{882}(89,\cdot)\) \(\chi_{882}(143,\cdot)\) \(\chi_{882}(269,\cdot)\) \(\chi_{882}(341,\cdot)\) \(\chi_{882}(395,\cdot)\) \(\chi_{882}(467,\cdot)\) \(\chi_{882}(593,\cdot)\) \(\chi_{882}(647,\cdot)\) \(\chi_{882}(719,\cdot)\) \(\chi_{882}(773,\cdot)\) \(\chi_{882}(845,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
Fixed field: | \(\Q(\zeta_{147})^+\) |
Values on generators
\((785,199)\) → \((-1,e\left(\frac{25}{42}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
\( \chi_{ 882 }(17, a) \) | \(1\) | \(1\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{21}\right)\) |
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)