from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(475, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([9,20]))
pari: [g,chi] = znchar(Mod(83,475))
Basic properties
| Modulus: | \(475\) | |
| Conductor: | \(475\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | 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.bd
\(\chi_{475}(83,\cdot)\) \(\chi_{475}(87,\cdot)\) \(\chi_{475}(102,\cdot)\) \(\chi_{475}(163,\cdot)\) \(\chi_{475}(178,\cdot)\) \(\chi_{475}(197,\cdot)\) \(\chi_{475}(258,\cdot)\) \(\chi_{475}(273,\cdot)\) \(\chi_{475}(277,\cdot)\) \(\chi_{475}(292,\cdot)\) \(\chi_{475}(353,\cdot)\) \(\chi_{475}(372,\cdot)\) \(\chi_{475}(387,\cdot)\) \(\chi_{475}(448,\cdot)\) \(\chi_{475}(463,\cdot)\) \(\chi_{475}(467,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((77,401)\) → \((e\left(\frac{3}{20}\right),e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 475 }(83, a) \) | \(-1\) | \(1\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(-i\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{31}{60}\right)\) |
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)