from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(799, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([0,29]))
pari: [g,chi] = znchar(Mod(120,799))
Basic properties
Modulus: | \(799\) | |
Conductor: | \(47\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(46\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{47}(26,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 799.l
\(\chi_{799}(35,\cdot)\) \(\chi_{799}(52,\cdot)\) \(\chi_{799}(69,\cdot)\) \(\chi_{799}(86,\cdot)\) \(\chi_{799}(120,\cdot)\) \(\chi_{799}(137,\cdot)\) \(\chi_{799}(154,\cdot)\) \(\chi_{799}(171,\cdot)\) \(\chi_{799}(273,\cdot)\) \(\chi_{799}(358,\cdot)\) \(\chi_{799}(409,\cdot)\) \(\chi_{799}(443,\cdot)\) \(\chi_{799}(511,\cdot)\) \(\chi_{799}(528,\cdot)\) \(\chi_{799}(562,\cdot)\) \(\chi_{799}(579,\cdot)\) \(\chi_{799}(630,\cdot)\) \(\chi_{799}(681,\cdot)\) \(\chi_{799}(698,\cdot)\) \(\chi_{799}(715,\cdot)\) \(\chi_{799}(749,\cdot)\) \(\chi_{799}(783,\cdot)\)
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: | Number field defined by a degree 46 polynomial |
Values on generators
\((377,52)\) → \((1,e\left(\frac{29}{46}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 799 }(120, a) \) | \(-1\) | \(1\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{29}{46}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{45}{46}\right)\) | \(e\left(\frac{19}{46}\right)\) |
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)