from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(89, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([35]))
pari: [g,chi] = znchar(Mod(5,89))
Basic properties
Modulus: | \(89\) | |
Conductor: | \(89\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | 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 89.g
\(\chi_{89}(5,\cdot)\) \(\chi_{89}(9,\cdot)\) \(\chi_{89}(10,\cdot)\) \(\chi_{89}(17,\cdot)\) \(\chi_{89}(18,\cdot)\) \(\chi_{89}(20,\cdot)\) \(\chi_{89}(21,\cdot)\) \(\chi_{89}(36,\cdot)\) \(\chi_{89}(40,\cdot)\) \(\chi_{89}(42,\cdot)\) \(\chi_{89}(47,\cdot)\) \(\chi_{89}(49,\cdot)\) \(\chi_{89}(53,\cdot)\) \(\chi_{89}(68,\cdot)\) \(\chi_{89}(69,\cdot)\) \(\chi_{89}(71,\cdot)\) \(\chi_{89}(72,\cdot)\) \(\chi_{89}(79,\cdot)\) \(\chi_{89}(80,\cdot)\) \(\chi_{89}(84,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
Fixed field: | Number field defined by a degree 44 polynomial |
Values on generators
\(3\) → \(e\left(\frac{35}{44}\right)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 89 }(5, a) \) | \(1\) | \(1\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) |
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)