from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(205, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([30,11]))
pari: [g,chi] = znchar(Mod(28,205))
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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 205.bb
\(\chi_{205}(13,\cdot)\) \(\chi_{205}(17,\cdot)\) \(\chi_{205}(22,\cdot)\) \(\chi_{205}(28,\cdot)\) \(\chi_{205}(47,\cdot)\) \(\chi_{205}(48,\cdot)\) \(\chi_{205}(53,\cdot)\) \(\chi_{205}(67,\cdot)\) \(\chi_{205}(93,\cdot)\) \(\chi_{205}(97,\cdot)\) \(\chi_{205}(117,\cdot)\) \(\chi_{205}(142,\cdot)\) \(\chi_{205}(147,\cdot)\) \(\chi_{205}(153,\cdot)\) \(\chi_{205}(193,\cdot)\) \(\chi_{205}(198,\cdot)\)
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: | 40.40.737344664758480175980241689422890987152982395783688920879217945970594882965087890625.1 |
Values on generators
\((42,6)\) → \((-i,e\left(\frac{11}{40}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 205 }(28, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(-i\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{31}{40}\right)\) |
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)