from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(243, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([31]))
pari: [g,chi] = znchar(Mod(35,243))
Basic properties
| Modulus: | \(243\) | |
| 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 \(\chi_{81}(65,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 243.h
\(\chi_{243}(8,\cdot)\) \(\chi_{243}(17,\cdot)\) \(\chi_{243}(35,\cdot)\) \(\chi_{243}(44,\cdot)\) \(\chi_{243}(62,\cdot)\) \(\chi_{243}(71,\cdot)\) \(\chi_{243}(89,\cdot)\) \(\chi_{243}(98,\cdot)\) \(\chi_{243}(116,\cdot)\) \(\chi_{243}(125,\cdot)\) \(\chi_{243}(143,\cdot)\) \(\chi_{243}(152,\cdot)\) \(\chi_{243}(170,\cdot)\) \(\chi_{243}(179,\cdot)\) \(\chi_{243}(197,\cdot)\) \(\chi_{243}(206,\cdot)\) \(\chi_{243}(224,\cdot)\) \(\chi_{243}(233,\cdot)\)
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
\(2\) → \(e\left(\frac{31}{54}\right)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 243 }(35, a) \) | \(-1\) | \(1\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{11}{54}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{25}{54}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{8}{27}\right)\) |
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)