from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(690, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,11,3]))
pari: [g,chi] = znchar(Mod(539,690))
Basic properties
Modulus: | \(690\) | |
Conductor: | \(345\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{345}(194,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 690.n
\(\chi_{690}(89,\cdot)\) \(\chi_{690}(149,\cdot)\) \(\chi_{690}(329,\cdot)\) \(\chi_{690}(359,\cdot)\) \(\chi_{690}(389,\cdot)\) \(\chi_{690}(419,\cdot)\) \(\chi_{690}(479,\cdot)\) \(\chi_{690}(539,\cdot)\) \(\chi_{690}(569,\cdot)\) \(\chi_{690}(659,\cdot)\)
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: | Number field defined by a degree 22 polynomial |
Values on generators
\((461,277,511)\) → \((-1,-1,e\left(\frac{3}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 690 }(539, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{2}{11}\right)\) |
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)