from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(572, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,12,25]))
pari: [g,chi] = znchar(Mod(279,572))
Basic properties
Modulus: | \(572\) | |
Conductor: | \(572\) | 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 572.bs
\(\chi_{572}(15,\cdot)\) \(\chi_{572}(59,\cdot)\) \(\chi_{572}(71,\cdot)\) \(\chi_{572}(115,\cdot)\) \(\chi_{572}(119,\cdot)\) \(\chi_{572}(163,\cdot)\) \(\chi_{572}(223,\cdot)\) \(\chi_{572}(267,\cdot)\) \(\chi_{572}(279,\cdot)\) \(\chi_{572}(323,\cdot)\) \(\chi_{572}(379,\cdot)\) \(\chi_{572}(383,\cdot)\) \(\chi_{572}(423,\cdot)\) \(\chi_{572}(427,\cdot)\) \(\chi_{572}(487,\cdot)\) \(\chi_{572}(531,\cdot)\)
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
\((287,365,353)\) → \((-1,e\left(\frac{1}{5}\right),e\left(\frac{5}{12}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) | \(25\) |
\( \chi_{ 572 }(279, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(i\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{10}\right)\) |
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)