from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(201, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,34]))
pari: [g,chi] = znchar(Mod(65,201))
Basic properties
Modulus: | \(201\) | |
Conductor: | \(201\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 201.o
\(\chi_{201}(17,\cdot)\) \(\chi_{201}(23,\cdot)\) \(\chi_{201}(26,\cdot)\) \(\chi_{201}(35,\cdot)\) \(\chi_{201}(47,\cdot)\) \(\chi_{201}(56,\cdot)\) \(\chi_{201}(65,\cdot)\) \(\chi_{201}(71,\cdot)\) \(\chi_{201}(77,\cdot)\) \(\chi_{201}(83,\cdot)\) \(\chi_{201}(86,\cdot)\) \(\chi_{201}(116,\cdot)\) \(\chi_{201}(122,\cdot)\) \(\chi_{201}(140,\cdot)\) \(\chi_{201}(155,\cdot)\) \(\chi_{201}(167,\cdot)\) \(\chi_{201}(170,\cdot)\) \(\chi_{201}(173,\cdot)\) \(\chi_{201}(188,\cdot)\) \(\chi_{201}(194,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
Fixed field: | Number field defined by a degree 66 polynomial |
Values on generators
\((68,136)\) → \((-1,e\left(\frac{17}{33}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 201 }(65, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{2}{33}\right)\) |
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)