from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(335, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,46]))
pari: [g,chi] = znchar(Mod(49,335))
Basic properties
| Modulus: | \(335\) | |
| Conductor: | \(335\) | 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 335.u
\(\chi_{335}(4,\cdot)\) \(\chi_{335}(19,\cdot)\) \(\chi_{335}(39,\cdot)\) \(\chi_{335}(49,\cdot)\) \(\chi_{335}(54,\cdot)\) \(\chi_{335}(84,\cdot)\) \(\chi_{335}(114,\cdot)\) \(\chi_{335}(144,\cdot)\) \(\chi_{335}(169,\cdot)\) \(\chi_{335}(189,\cdot)\) \(\chi_{335}(194,\cdot)\) \(\chi_{335}(199,\cdot)\) \(\chi_{335}(224,\cdot)\) \(\chi_{335}(234,\cdot)\) \(\chi_{335}(274,\cdot)\) \(\chi_{335}(284,\cdot)\) \(\chi_{335}(289,\cdot)\) \(\chi_{335}(294,\cdot)\) \(\chi_{335}(304,\cdot)\) \(\chi_{335}(324,\cdot)\)
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
\((202,136)\) → \((-1,e\left(\frac{23}{33}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 335 }(49, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{49}{66}\right)\) |
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)