from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(139, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([12]))
pari: [g,chi] = znchar(Mod(100,139))
Basic properties
| Modulus: | \(139\) | |
| Conductor: | \(139\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(23\) | 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 139.e
\(\chi_{139}(6,\cdot)\) \(\chi_{139}(34,\cdot)\) \(\chi_{139}(36,\cdot)\) \(\chi_{139}(44,\cdot)\) \(\chi_{139}(45,\cdot)\) \(\chi_{139}(52,\cdot)\) \(\chi_{139}(55,\cdot)\) \(\chi_{139}(57,\cdot)\) \(\chi_{139}(63,\cdot)\) \(\chi_{139}(64,\cdot)\) \(\chi_{139}(65,\cdot)\) \(\chi_{139}(77,\cdot)\) \(\chi_{139}(79,\cdot)\) \(\chi_{139}(80,\cdot)\) \(\chi_{139}(91,\cdot)\) \(\chi_{139}(100,\cdot)\) \(\chi_{139}(106,\cdot)\) \(\chi_{139}(112,\cdot)\) \(\chi_{139}(116,\cdot)\) \(\chi_{139}(125,\cdot)\) \(\chi_{139}(129,\cdot)\) \(\chi_{139}(131,\cdot)\)
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: | \(\Q(\zeta_{23})\) |
| Fixed field: | Number field defined by a degree 23 polynomial |
Values on generators
\(2\) → \(e\left(\frac{6}{23}\right)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 139 }(100, a) \) | \(1\) | \(1\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) |
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)