from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(799, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([23,1]))
pari: [g,chi] = znchar(Mod(475,799))
Basic properties
Modulus: | \(799\) | |
Conductor: | \(799\) | 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 799.m
\(\chi_{799}(33,\cdot)\) \(\chi_{799}(67,\cdot)\) \(\chi_{799}(135,\cdot)\) \(\chi_{799}(152,\cdot)\) \(\chi_{799}(186,\cdot)\) \(\chi_{799}(203,\cdot)\) \(\chi_{799}(254,\cdot)\) \(\chi_{799}(305,\cdot)\) \(\chi_{799}(322,\cdot)\) \(\chi_{799}(339,\cdot)\) \(\chi_{799}(373,\cdot)\) \(\chi_{799}(407,\cdot)\) \(\chi_{799}(458,\cdot)\) \(\chi_{799}(475,\cdot)\) \(\chi_{799}(492,\cdot)\) \(\chi_{799}(509,\cdot)\) \(\chi_{799}(543,\cdot)\) \(\chi_{799}(560,\cdot)\) \(\chi_{799}(577,\cdot)\) \(\chi_{799}(594,\cdot)\) \(\chi_{799}(696,\cdot)\) \(\chi_{799}(781,\cdot)\)
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: | 46.0.35053291047098822445710921889723167962319821671472348329989416363445958210915442479511871796900053441791.1 |
Values on generators
\((377,52)\) → \((-1,e\left(\frac{1}{46}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 799 }(475, a) \) | \(-1\) | \(1\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{43}{46}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{15}{46}\right)\) | \(e\left(\frac{9}{46}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) |
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)