from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(975, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,3,5]))
pari: [g,chi] = znchar(Mod(2,975))
Basic properties
| Modulus: | \(975\) | |
| Conductor: | \(975\) | 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 975.cr
\(\chi_{975}(2,\cdot)\) \(\chi_{975}(98,\cdot)\) \(\chi_{975}(128,\cdot)\) \(\chi_{975}(197,\cdot)\) \(\chi_{975}(227,\cdot)\) \(\chi_{975}(323,\cdot)\) \(\chi_{975}(392,\cdot)\) \(\chi_{975}(422,\cdot)\) \(\chi_{975}(488,\cdot)\) \(\chi_{975}(587,\cdot)\) \(\chi_{975}(617,\cdot)\) \(\chi_{975}(683,\cdot)\) \(\chi_{975}(713,\cdot)\) \(\chi_{975}(812,\cdot)\) \(\chi_{975}(878,\cdot)\) \(\chi_{975}(908,\cdot)\)
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
\((326,352,301)\) → \((-1,e\left(\frac{1}{20}\right),e\left(\frac{1}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
| \( \chi_{ 975 }(2, a) \) | \(-1\) | \(1\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{31}{60}\right)\) |
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)