from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(169, base_ring=CyclotomicField(52))
M = H._module
chi = DirichletCharacter(H, M([7]))
pari: [g,chi] = znchar(Mod(31,169))
Basic properties
Modulus: | \(169\) | |
Conductor: | \(169\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(52\) | 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 169.j
\(\chi_{169}(5,\cdot)\) \(\chi_{169}(8,\cdot)\) \(\chi_{169}(18,\cdot)\) \(\chi_{169}(21,\cdot)\) \(\chi_{169}(31,\cdot)\) \(\chi_{169}(34,\cdot)\) \(\chi_{169}(44,\cdot)\) \(\chi_{169}(47,\cdot)\) \(\chi_{169}(57,\cdot)\) \(\chi_{169}(60,\cdot)\) \(\chi_{169}(73,\cdot)\) \(\chi_{169}(83,\cdot)\) \(\chi_{169}(86,\cdot)\) \(\chi_{169}(96,\cdot)\) \(\chi_{169}(109,\cdot)\) \(\chi_{169}(112,\cdot)\) \(\chi_{169}(122,\cdot)\) \(\chi_{169}(125,\cdot)\) \(\chi_{169}(135,\cdot)\) \(\chi_{169}(138,\cdot)\) \(\chi_{169}(148,\cdot)\) \(\chi_{169}(151,\cdot)\) \(\chi_{169}(161,\cdot)\) \(\chi_{169}(164,\cdot)\)
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: | $\Q(\zeta_{52})$ |
Fixed field: | Number field defined by a degree 52 polynomial |
Values on generators
\(2\) → \(e\left(\frac{7}{52}\right)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 169 }(31, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{52}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{11}{52}\right)\) | \(e\left(\frac{43}{52}\right)\) | \(e\left(\frac{21}{52}\right)\) | \(e\left(\frac{21}{52}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{45}{52}\right)\) |
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)