from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(666, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([24,1]))
pari: [g,chi] = znchar(Mod(187,666))
Basic properties
| Modulus: | \(666\) | |
| Conductor: | \(333\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{333}(187,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 666.bq
\(\chi_{666}(13,\cdot)\) \(\chi_{666}(133,\cdot)\) \(\chi_{666}(187,\cdot)\) \(\chi_{666}(205,\cdot)\) \(\chi_{666}(283,\cdot)\) \(\chi_{666}(301,\cdot)\) \(\chi_{666}(313,\cdot)\) \(\chi_{666}(331,\cdot)\) \(\chi_{666}(385,\cdot)\) \(\chi_{666}(463,\cdot)\) \(\chi_{666}(499,\cdot)\) \(\chi_{666}(661,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
| Fixed field: | Number field defined by a degree 36 polynomial |
Values on generators
\((371,631)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{1}{36}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 666 }(187, a) \) | \(-1\) | \(1\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(-1\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(-i\) | \(e\left(\frac{17}{18}\right)\) | \(i\) | \(e\left(\frac{7}{12}\right)\) |
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)