from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(106, base_ring=CyclotomicField(52))
M = H._module
chi = DirichletCharacter(H, M([17]))
pari: [g,chi] = znchar(Mod(3,106))
Basic properties
| Modulus: | \(106\) | |
| Conductor: | \(53\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(52\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{53}(3,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 106.f
\(\chi_{106}(3,\cdot)\) \(\chi_{106}(5,\cdot)\) \(\chi_{106}(19,\cdot)\) \(\chi_{106}(21,\cdot)\) \(\chi_{106}(27,\cdot)\) \(\chi_{106}(31,\cdot)\) \(\chi_{106}(33,\cdot)\) \(\chi_{106}(35,\cdot)\) \(\chi_{106}(39,\cdot)\) \(\chi_{106}(41,\cdot)\) \(\chi_{106}(45,\cdot)\) \(\chi_{106}(51,\cdot)\) \(\chi_{106}(55,\cdot)\) \(\chi_{106}(61,\cdot)\) \(\chi_{106}(65,\cdot)\) \(\chi_{106}(67,\cdot)\) \(\chi_{106}(71,\cdot)\) \(\chi_{106}(73,\cdot)\) \(\chi_{106}(75,\cdot)\) \(\chi_{106}(79,\cdot)\) \(\chi_{106}(85,\cdot)\) \(\chi_{106}(87,\cdot)\) \(\chi_{106}(101,\cdot)\) \(\chi_{106}(103,\cdot)\)
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
\(55\) → \(e\left(\frac{17}{52}\right)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 106 }(3, a) \) | \(-1\) | \(1\) | \(e\left(\frac{29}{52}\right)\) | \(e\left(\frac{19}{52}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{5}{52}\right)\) | \(e\left(\frac{7}{52}\right)\) |
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)