from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(975, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,42,35]))
pari: [g,chi] = znchar(Mod(284,975))
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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 975.cw
\(\chi_{975}(59,\cdot)\) \(\chi_{975}(89,\cdot)\) \(\chi_{975}(119,\cdot)\) \(\chi_{975}(254,\cdot)\) \(\chi_{975}(284,\cdot)\) \(\chi_{975}(314,\cdot)\) \(\chi_{975}(344,\cdot)\) \(\chi_{975}(479,\cdot)\) \(\chi_{975}(509,\cdot)\) \(\chi_{975}(539,\cdot)\) \(\chi_{975}(644,\cdot)\) \(\chi_{975}(704,\cdot)\) \(\chi_{975}(734,\cdot)\) \(\chi_{975}(839,\cdot)\) \(\chi_{975}(869,\cdot)\) \(\chi_{975}(929,\cdot)\)
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
\((326,352,301)\) → \((-1,e\left(\frac{7}{10}\right),e\left(\frac{7}{12}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
\( \chi_{ 975 }(284, a) \) | \(1\) | \(1\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) |
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)