from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(119, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([16,3]))
pari: [g,chi] = znchar(Mod(37,119))
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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 119.t
\(\chi_{119}(11,\cdot)\) \(\chi_{119}(23,\cdot)\) \(\chi_{119}(37,\cdot)\) \(\chi_{119}(39,\cdot)\) \(\chi_{119}(44,\cdot)\) \(\chi_{119}(46,\cdot)\) \(\chi_{119}(58,\cdot)\) \(\chi_{119}(65,\cdot)\) \(\chi_{119}(74,\cdot)\) \(\chi_{119}(79,\cdot)\) \(\chi_{119}(88,\cdot)\) \(\chi_{119}(95,\cdot)\) \(\chi_{119}(107,\cdot)\) \(\chi_{119}(109,\cdot)\) \(\chi_{119}(114,\cdot)\) \(\chi_{119}(116,\cdot)\)
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
\((52,71)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{1}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 119 }(37, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{23}{48}\right)\) |
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)