from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(81, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([26]))
pari: [g,chi] = znchar(Mod(40,81))
Basic properties
| Modulus: | \(81\) | |
| Conductor: | \(81\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(27\) | 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 81.g
\(\chi_{81}(4,\cdot)\) \(\chi_{81}(7,\cdot)\) \(\chi_{81}(13,\cdot)\) \(\chi_{81}(16,\cdot)\) \(\chi_{81}(22,\cdot)\) \(\chi_{81}(25,\cdot)\) \(\chi_{81}(31,\cdot)\) \(\chi_{81}(34,\cdot)\) \(\chi_{81}(40,\cdot)\) \(\chi_{81}(43,\cdot)\) \(\chi_{81}(49,\cdot)\) \(\chi_{81}(52,\cdot)\) \(\chi_{81}(58,\cdot)\) \(\chi_{81}(61,\cdot)\) \(\chi_{81}(67,\cdot)\) \(\chi_{81}(70,\cdot)\) \(\chi_{81}(76,\cdot)\) \(\chi_{81}(79,\cdot)\)
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 27 polynomial |
Values on generators
\(2\) → \(e\left(\frac{13}{27}\right)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 81 }(40, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{25}{27}\right)\) |
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)