from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(161, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([11,27]))
pari: [g,chi] = znchar(Mod(80,161))
Basic properties
| Modulus: | \(161\) | |
| Conductor: | \(161\) | 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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 161.o
\(\chi_{161}(5,\cdot)\) \(\chi_{161}(10,\cdot)\) \(\chi_{161}(17,\cdot)\) \(\chi_{161}(19,\cdot)\) \(\chi_{161}(33,\cdot)\) \(\chi_{161}(38,\cdot)\) \(\chi_{161}(40,\cdot)\) \(\chi_{161}(61,\cdot)\) \(\chi_{161}(66,\cdot)\) \(\chi_{161}(80,\cdot)\) \(\chi_{161}(89,\cdot)\) \(\chi_{161}(103,\cdot)\) \(\chi_{161}(122,\cdot)\) \(\chi_{161}(129,\cdot)\) \(\chi_{161}(136,\cdot)\) \(\chi_{161}(143,\cdot)\) \(\chi_{161}(145,\cdot)\) \(\chi_{161}(152,\cdot)\) \(\chi_{161}(157,\cdot)\) \(\chi_{161}(159,\cdot)\)
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
\((24,120)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{9}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 161 }(80, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{1}{66}\right)\) |
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)