from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(799, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([0,12]))
pari: [g,chi] = znchar(Mod(18,799))
Basic properties
| Modulus: | \(799\) | |
| Conductor: | \(47\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(23\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{47}(18,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 799.k
\(\chi_{799}(18,\cdot)\) \(\chi_{799}(103,\cdot)\) \(\chi_{799}(205,\cdot)\) \(\chi_{799}(222,\cdot)\) \(\chi_{799}(239,\cdot)\) \(\chi_{799}(256,\cdot)\) \(\chi_{799}(290,\cdot)\) \(\chi_{799}(307,\cdot)\) \(\chi_{799}(324,\cdot)\) \(\chi_{799}(341,\cdot)\) \(\chi_{799}(392,\cdot)\) \(\chi_{799}(426,\cdot)\) \(\chi_{799}(460,\cdot)\) \(\chi_{799}(477,\cdot)\) \(\chi_{799}(494,\cdot)\) \(\chi_{799}(545,\cdot)\) \(\chi_{799}(596,\cdot)\) \(\chi_{799}(613,\cdot)\) \(\chi_{799}(647,\cdot)\) \(\chi_{799}(664,\cdot)\) \(\chi_{799}(732,\cdot)\) \(\chi_{799}(766,\cdot)\)
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 23 polynomial |
Values on generators
\((377,52)\) → \((1,e\left(\frac{6}{23}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 799 }(18, a) \) | \(1\) | \(1\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) |
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)