from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(106, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([7]))
pari: [g,chi] = znchar(Mod(7,106))
Basic properties
| Modulus: | \(106\) | |
| Conductor: | \(53\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(26\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{53}(7,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 106.e
\(\chi_{106}(7,\cdot)\) \(\chi_{106}(9,\cdot)\) \(\chi_{106}(11,\cdot)\) \(\chi_{106}(17,\cdot)\) \(\chi_{106}(25,\cdot)\) \(\chi_{106}(29,\cdot)\) \(\chi_{106}(37,\cdot)\) \(\chi_{106}(43,\cdot)\) \(\chi_{106}(57,\cdot)\) \(\chi_{106}(59,\cdot)\) \(\chi_{106}(91,\cdot)\) \(\chi_{106}(93,\cdot)\)
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: | \(\Q(\zeta_{13})\) |
| Fixed field: | Number field defined by a degree 26 polynomial |
Values on generators
\(55\) → \(e\left(\frac{7}{26}\right)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 106 }(7, a) \) | \(1\) | \(1\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{9}{26}\right)\) |
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)