from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(456, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([0,0,9,5]))
pari: [g,chi] = znchar(Mod(89,456))
Basic properties
| Modulus: | \(456\) | |
| Conductor: | \(57\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(18\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{57}(32,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 456.bm
\(\chi_{456}(41,\cdot)\) \(\chi_{456}(89,\cdot)\) \(\chi_{456}(185,\cdot)\) \(\chi_{456}(257,\cdot)\) \(\chi_{456}(281,\cdot)\) \(\chi_{456}(401,\cdot)\)
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: | \(\Q(\zeta_{57})^+\) |
Values on generators
\((343,229,305,97)\) → \((1,1,-1,e\left(\frac{5}{18}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 456 }(89, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{18}\right)\) |
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)