from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(181, base_ring=CyclotomicField(90))
M = H._module
chi = DirichletCharacter(H, M([7]))
pari: [g,chi] = znchar(Mod(94,181))
Basic properties
Modulus: | \(181\) | |
Conductor: | \(181\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(90\) | 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 181.q
\(\chi_{181}(4,\cdot)\) \(\chi_{181}(11,\cdot)\) \(\chi_{181}(12,\cdot)\) \(\chi_{181}(20,\cdot)\) \(\chi_{181}(33,\cdot)\) \(\chi_{181}(37,\cdot)\) \(\chi_{181}(52,\cdot)\) \(\chi_{181}(55,\cdot)\) \(\chi_{181}(60,\cdot)\) \(\chi_{181}(79,\cdot)\) \(\chi_{181}(94,\cdot)\) \(\chi_{181}(100,\cdot)\) \(\chi_{181}(106,\cdot)\) \(\chi_{181}(111,\cdot)\) \(\chi_{181}(136,\cdot)\) \(\chi_{181}(137,\cdot)\) \(\chi_{181}(143,\cdot)\) \(\chi_{181}(147,\cdot)\) \(\chi_{181}(165,\cdot)\) \(\chi_{181}(166,\cdot)\) \(\chi_{181}(167,\cdot)\) \(\chi_{181}(168,\cdot)\) \(\chi_{181}(172,\cdot)\) \(\chi_{181}(178,\cdot)\)
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: | $\Q(\zeta_{45})$ |
Fixed field: | Number field defined by a degree 90 polynomial |
Values on generators
\(2\) → \(e\left(\frac{7}{90}\right)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 181 }(94, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{90}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{19}{90}\right)\) | \(e\left(\frac{37}{45}\right)\) |
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)