from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(955, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([19,20]))
pari: [g,chi] = znchar(Mod(789,955))
Basic properties
Modulus: | \(955\) | |
Conductor: | \(955\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(38\) | 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 955.o
\(\chi_{955}(69,\cdot)\) \(\chi_{955}(154,\cdot)\) \(\chi_{955}(344,\cdot)\) \(\chi_{955}(414,\cdot)\) \(\chi_{955}(434,\cdot)\) \(\chi_{955}(489,\cdot)\) \(\chi_{955}(559,\cdot)\) \(\chi_{955}(579,\cdot)\) \(\chi_{955}(609,\cdot)\) \(\chi_{955}(694,\cdot)\) \(\chi_{955}(709,\cdot)\) \(\chi_{955}(769,\cdot)\) \(\chi_{955}(789,\cdot)\) \(\chi_{955}(794,\cdot)\) \(\chi_{955}(889,\cdot)\) \(\chi_{955}(914,\cdot)\) \(\chi_{955}(924,\cdot)\) \(\chi_{955}(944,\cdot)\)
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: | \(\Q(\zeta_{19})\) |
Fixed field: | Number field defined by a degree 38 polynomial |
Values on generators
\((192,401)\) → \((-1,e\left(\frac{10}{19}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
\( \chi_{ 955 }(789, a) \) | \(1\) | \(1\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{21}{38}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(-1\) | \(e\left(\frac{37}{38}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{17}{38}\right)\) |
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)