from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(235, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([23,9]))
pari: [g,chi] = znchar(Mod(134,235))
Basic properties
Modulus: | \(235\) | |
Conductor: | \(235\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(46\) | 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 235.j
\(\chi_{235}(19,\cdot)\) \(\chi_{235}(29,\cdot)\) \(\chi_{235}(39,\cdot)\) \(\chi_{235}(44,\cdot)\) \(\chi_{235}(69,\cdot)\) \(\chi_{235}(99,\cdot)\) \(\chi_{235}(104,\cdot)\) \(\chi_{235}(109,\cdot)\) \(\chi_{235}(114,\cdot)\) \(\chi_{235}(124,\cdot)\) \(\chi_{235}(129,\cdot)\) \(\chi_{235}(134,\cdot)\) \(\chi_{235}(139,\cdot)\) \(\chi_{235}(154,\cdot)\) \(\chi_{235}(164,\cdot)\) \(\chi_{235}(174,\cdot)\) \(\chi_{235}(179,\cdot)\) \(\chi_{235}(184,\cdot)\) \(\chi_{235}(199,\cdot)\) \(\chi_{235}(214,\cdot)\) \(\chi_{235}(219,\cdot)\) \(\chi_{235}(229,\cdot)\)
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: | \(\Q(\zeta_{23})\) |
Fixed field: | 46.0.20927324417353262576794873573459132405730192352302940670222843367282253010356426239013671875.1 |
Values on generators
\((142,146)\) → \((-1,e\left(\frac{9}{46}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
\( \chi_{ 235 }(134, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{46}\right)\) | \(e\left(\frac{19}{46}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{35}{46}\right)\) | \(e\left(\frac{3}{46}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{17}{46}\right)\) | \(e\left(\frac{21}{46}\right)\) | \(e\left(\frac{15}{23}\right)\) |
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)