from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(803, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([0,2]))
pari: [g,chi] = znchar(Mod(221,803))
Basic properties
Modulus: | \(803\) | |
Conductor: | \(73\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(9\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{73}(2,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 803.n
\(\chi_{803}(89,\cdot)\) \(\chi_{803}(221,\cdot)\) \(\chi_{803}(397,\cdot)\) \(\chi_{803}(639,\cdot)\) \(\chi_{803}(661,\cdot)\) \(\chi_{803}(694,\cdot)\)
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: | \(\Q(\zeta_{9})\) |
Fixed field: | 9.9.806460091894081.1 |
Values on generators
\((585,78)\) → \((1,e\left(\frac{1}{9}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
\( \chi_{ 803 }(221, a) \) | \(1\) | \(1\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(1\) | \(e\left(\frac{4}{9}\right)\) |
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)