from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(475, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([9,5]))
pari: [g,chi] = znchar(Mod(464,475))
Basic properties
| Modulus: | \(475\) | |
| Conductor: | \(475\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | 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 475.z
\(\chi_{475}(69,\cdot)\) \(\chi_{475}(84,\cdot)\) \(\chi_{475}(164,\cdot)\) \(\chi_{475}(179,\cdot)\) \(\chi_{475}(259,\cdot)\) \(\chi_{475}(354,\cdot)\) \(\chi_{475}(369,\cdot)\) \(\chi_{475}(464,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
| Fixed field: | 30.0.41334267425810406820389890772071250779617912485264241695404052734375.1 |
Values on generators
\((77,401)\) → \((e\left(\frac{3}{10}\right),e\left(\frac{1}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 475 }(464, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(-1\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{8}{15}\right)\) |
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)