from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(169, base_ring=CyclotomicField(78))
M = H._module
chi = DirichletCharacter(H, M([25]))
pari: [g,chi] = znchar(Mod(121,169))
Basic properties
| Modulus: | \(169\) | |
| Conductor: | \(169\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(78\) | 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 169.k
\(\chi_{169}(4,\cdot)\) \(\chi_{169}(10,\cdot)\) \(\chi_{169}(17,\cdot)\) \(\chi_{169}(30,\cdot)\) \(\chi_{169}(36,\cdot)\) \(\chi_{169}(43,\cdot)\) \(\chi_{169}(49,\cdot)\) \(\chi_{169}(56,\cdot)\) \(\chi_{169}(62,\cdot)\) \(\chi_{169}(69,\cdot)\) \(\chi_{169}(75,\cdot)\) \(\chi_{169}(82,\cdot)\) \(\chi_{169}(88,\cdot)\) \(\chi_{169}(95,\cdot)\) \(\chi_{169}(101,\cdot)\) \(\chi_{169}(108,\cdot)\) \(\chi_{169}(114,\cdot)\) \(\chi_{169}(121,\cdot)\) \(\chi_{169}(127,\cdot)\) \(\chi_{169}(134,\cdot)\) \(\chi_{169}(140,\cdot)\) \(\chi_{169}(153,\cdot)\) \(\chi_{169}(160,\cdot)\) \(\chi_{169}(166,\cdot)\)
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: | $\Q(\zeta_{39})$ |
| Fixed field: | Number field defined by a degree 78 polynomial |
Values on generators
\(2\) → \(e\left(\frac{25}{78}\right)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 169 }(121, a) \) | \(1\) | \(1\) | \(e\left(\frac{25}{78}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{5}{78}\right)\) | \(e\left(\frac{23}{78}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{1}{78}\right)\) |
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)