from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(775, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([27,8]))
pari: [g,chi] = znchar(Mod(112,775))
Basic properties
Modulus: | \(775\) | |
Conductor: | \(775\) | 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 775.db
\(\chi_{775}(112,\cdot)\) \(\chi_{775}(152,\cdot)\) \(\chi_{775}(173,\cdot)\) \(\chi_{775}(317,\cdot)\) \(\chi_{775}(392,\cdot)\) \(\chi_{775}(413,\cdot)\) \(\chi_{775}(452,\cdot)\) \(\chi_{775}(537,\cdot)\) \(\chi_{775}(572,\cdot)\) \(\chi_{775}(603,\cdot)\) \(\chi_{775}(648,\cdot)\) \(\chi_{775}(658,\cdot)\) \(\chi_{775}(722,\cdot)\) \(\chi_{775}(733,\cdot)\) \(\chi_{775}(753,\cdot)\) \(\chi_{775}(763,\cdot)\)
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
\((652,251)\) → \((e\left(\frac{9}{20}\right),e\left(\frac{2}{15}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
\( \chi_{ 775 }(112, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{60}\right)\) |
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)