from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(207, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([55,24]))
pari: [g,chi] = znchar(Mod(131,207))
Basic properties
| Modulus: | \(207\) | |
| Conductor: | \(207\) | 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 207.n
\(\chi_{207}(2,\cdot)\) \(\chi_{207}(29,\cdot)\) \(\chi_{207}(32,\cdot)\) \(\chi_{207}(41,\cdot)\) \(\chi_{207}(50,\cdot)\) \(\chi_{207}(59,\cdot)\) \(\chi_{207}(77,\cdot)\) \(\chi_{207}(95,\cdot)\) \(\chi_{207}(101,\cdot)\) \(\chi_{207}(104,\cdot)\) \(\chi_{207}(110,\cdot)\) \(\chi_{207}(119,\cdot)\) \(\chi_{207}(128,\cdot)\) \(\chi_{207}(131,\cdot)\) \(\chi_{207}(140,\cdot)\) \(\chi_{207}(146,\cdot)\) \(\chi_{207}(164,\cdot)\) \(\chi_{207}(167,\cdot)\) \(\chi_{207}(173,\cdot)\) \(\chi_{207}(200,\cdot)\)
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
\((47,28)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{4}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 207 }(131, a) \) | \(-1\) | \(1\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{8}{33}\right)\) |
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)