from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(205, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,13]))
pari: [g,chi] = znchar(Mod(24,205))
Basic properties
| Modulus: | \(205\) | |
| Conductor: | \(205\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | 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 205.ba
\(\chi_{205}(19,\cdot)\) \(\chi_{205}(24,\cdot)\) \(\chi_{205}(29,\cdot)\) \(\chi_{205}(34,\cdot)\) \(\chi_{205}(54,\cdot)\) \(\chi_{205}(69,\cdot)\) \(\chi_{205}(89,\cdot)\) \(\chi_{205}(94,\cdot)\) \(\chi_{205}(99,\cdot)\) \(\chi_{205}(104,\cdot)\) \(\chi_{205}(129,\cdot)\) \(\chi_{205}(134,\cdot)\) \(\chi_{205}(149,\cdot)\) \(\chi_{205}(179,\cdot)\) \(\chi_{205}(194,\cdot)\) \(\chi_{205}(199,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
| Fixed field: | 40.0.75504093671268370020376748996904037084465397328249745498031917667388916015625.1 |
Values on generators
\((42,6)\) → \((-1,e\left(\frac{13}{40}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 205 }(24, a) \) | \(-1\) | \(1\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(-i\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) |
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)