from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(162, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([13]))
pari: [g,chi] = znchar(Mod(11,162))
Basic properties
| Modulus: | \(162\) | |
| Conductor: | \(81\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{81}(11,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 162.h
\(\chi_{162}(5,\cdot)\) \(\chi_{162}(11,\cdot)\) \(\chi_{162}(23,\cdot)\) \(\chi_{162}(29,\cdot)\) \(\chi_{162}(41,\cdot)\) \(\chi_{162}(47,\cdot)\) \(\chi_{162}(59,\cdot)\) \(\chi_{162}(65,\cdot)\) \(\chi_{162}(77,\cdot)\) \(\chi_{162}(83,\cdot)\) \(\chi_{162}(95,\cdot)\) \(\chi_{162}(101,\cdot)\) \(\chi_{162}(113,\cdot)\) \(\chi_{162}(119,\cdot)\) \(\chi_{162}(131,\cdot)\) \(\chi_{162}(137,\cdot)\) \(\chi_{162}(149,\cdot)\) \(\chi_{162}(155,\cdot)\)
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: | \(\Q(\zeta_{27})\) |
| Fixed field: | Number field defined by a degree 54 polynomial |
Values on generators
\(83\) → \(e\left(\frac{13}{54}\right)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 162 }(11, a) \) | \(-1\) | \(1\) | \(e\left(\frac{29}{54}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{7}{54}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{22}{27}\right)\) |
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)