from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(119, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([8,9]))
pari: [g,chi] = znchar(Mod(100,119))
Basic properties
| Modulus: | \(119\) | |
| Conductor: | \(119\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(24\) | 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 119.q
\(\chi_{119}(2,\cdot)\) \(\chi_{119}(9,\cdot)\) \(\chi_{119}(25,\cdot)\) \(\chi_{119}(32,\cdot)\) \(\chi_{119}(53,\cdot)\) \(\chi_{119}(60,\cdot)\) \(\chi_{119}(93,\cdot)\) \(\chi_{119}(100,\cdot)\)
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: | \(\Q(\zeta_{24})\) |
| Fixed field: | 24.24.2296127442650479958000916502307873630417.1 |
Values on generators
\((52,71)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{3}{8}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 119 }(100, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(-i\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{13}{24}\right)\) |
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)