from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(517, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([23,19]))
pari: [g,chi] = znchar(Mod(10,517))
Basic properties
| Modulus: | \(517\) | |
| Conductor: | \(517\) | 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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 517.l
\(\chi_{517}(10,\cdot)\) \(\chi_{517}(43,\cdot)\) \(\chi_{517}(76,\cdot)\) \(\chi_{517}(87,\cdot)\) \(\chi_{517}(109,\cdot)\) \(\chi_{517}(120,\cdot)\) \(\chi_{517}(164,\cdot)\) \(\chi_{517}(186,\cdot)\) \(\chi_{517}(208,\cdot)\) \(\chi_{517}(219,\cdot)\) \(\chi_{517}(274,\cdot)\) \(\chi_{517}(340,\cdot)\) \(\chi_{517}(351,\cdot)\) \(\chi_{517}(362,\cdot)\) \(\chi_{517}(373,\cdot)\) \(\chi_{517}(395,\cdot)\) \(\chi_{517}(406,\cdot)\) \(\chi_{517}(417,\cdot)\) \(\chi_{517}(428,\cdot)\) \(\chi_{517}(461,\cdot)\) \(\chi_{517}(483,\cdot)\) \(\chi_{517}(505,\cdot)\)
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: | 46.46.1571937830040451992228674568491756687115540075866136861286122171472474917522743511699927590412842317.1 |
Values on generators
\((189,287)\) → \((-1,e\left(\frac{19}{46}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 517 }(10, a) \) | \(1\) | \(1\) | \(e\left(\frac{43}{46}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{19}{46}\right)\) | \(e\left(\frac{9}{46}\right)\) | \(e\left(\frac{33}{46}\right)\) | \(e\left(\frac{37}{46}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) |
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)