from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(119, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([40,15]))
pari: [g,chi] = znchar(Mod(5,119))
Basic properties
| Modulus: | \(119\) | |
| Conductor: | \(119\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | 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.s
\(\chi_{119}(3,\cdot)\) \(\chi_{119}(5,\cdot)\) \(\chi_{119}(10,\cdot)\) \(\chi_{119}(12,\cdot)\) \(\chi_{119}(24,\cdot)\) \(\chi_{119}(31,\cdot)\) \(\chi_{119}(40,\cdot)\) \(\chi_{119}(45,\cdot)\) \(\chi_{119}(54,\cdot)\) \(\chi_{119}(61,\cdot)\) \(\chi_{119}(73,\cdot)\) \(\chi_{119}(75,\cdot)\) \(\chi_{119}(80,\cdot)\) \(\chi_{119}(82,\cdot)\) \(\chi_{119}(96,\cdot)\) \(\chi_{119}(108,\cdot)\)
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: | \(\Q(\zeta_{48})\) |
| Fixed field: | Number field defined by a degree 48 polynomial |
Values on generators
\((52,71)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{5}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 119 }(5, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{11}{48}\right)\) |
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)