from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(111, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([0,5]))
pari: [g,chi] = znchar(Mod(97,111))
Basic properties
| Modulus: | \(111\) | |
| Conductor: | \(37\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(12\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{37}(23,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 111.l
\(\chi_{111}(82,\cdot)\) \(\chi_{111}(88,\cdot)\) \(\chi_{111}(97,\cdot)\) \(\chi_{111}(103,\cdot)\)
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: | \(\Q(\zeta_{12})\) |
| Fixed field: | 12.0.177917621779460413.1 |
Values on generators
\((38,76)\) → \((1,e\left(\frac{5}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 111 }(97, a) \) | \(-1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(i\) | \(1\) | \(-1\) | \(e\left(\frac{7}{12}\right)\) | \(-i\) | \(e\left(\frac{2}{3}\right)\) |
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)