from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(165, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,15,2]))
pari: [g,chi] = znchar(Mod(68,165))
Basic properties
| Modulus: | \(165\) | |
| Conductor: | \(165\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | 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 165.u
\(\chi_{165}(2,\cdot)\) \(\chi_{165}(8,\cdot)\) \(\chi_{165}(17,\cdot)\) \(\chi_{165}(62,\cdot)\) \(\chi_{165}(68,\cdot)\) \(\chi_{165}(83,\cdot)\) \(\chi_{165}(107,\cdot)\) \(\chi_{165}(128,\cdot)\)
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: | \(\Q(\zeta_{20})\) |
| Fixed field: | Number field defined by a degree 20 polynomial |
Values on generators
\((56,67,46)\) → \((-1,-i,e\left(\frac{1}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(23\) |
| \( \chi_{ 165 }(68, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(-i\) |
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)