from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(243, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([52]))
pari: [g,chi] = znchar(Mod(19,243))
Basic properties
| Modulus: | \(243\) | |
| Conductor: | \(81\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(27\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{81}(61,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 243.g
\(\chi_{243}(10,\cdot)\) \(\chi_{243}(19,\cdot)\) \(\chi_{243}(37,\cdot)\) \(\chi_{243}(46,\cdot)\) \(\chi_{243}(64,\cdot)\) \(\chi_{243}(73,\cdot)\) \(\chi_{243}(91,\cdot)\) \(\chi_{243}(100,\cdot)\) \(\chi_{243}(118,\cdot)\) \(\chi_{243}(127,\cdot)\) \(\chi_{243}(145,\cdot)\) \(\chi_{243}(154,\cdot)\) \(\chi_{243}(172,\cdot)\) \(\chi_{243}(181,\cdot)\) \(\chi_{243}(199,\cdot)\) \(\chi_{243}(208,\cdot)\) \(\chi_{243}(226,\cdot)\) \(\chi_{243}(235,\cdot)\)
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
\(2\) → \(e\left(\frac{26}{27}\right)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 243 }(19, a) \) | \(1\) | \(1\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{23}{27}\right)\) |
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)