from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(629, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([9,11]))
pari: [g,chi] = znchar(Mod(13,629))
Basic properties
| Modulus: | \(629\) | |
| Conductor: | \(629\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | 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 629.bp
\(\chi_{629}(13,\cdot)\) \(\chi_{629}(55,\cdot)\) \(\chi_{629}(89,\cdot)\) \(\chi_{629}(106,\cdot)\) \(\chi_{629}(183,\cdot)\) \(\chi_{629}(242,\cdot)\) \(\chi_{629}(387,\cdot)\) \(\chi_{629}(446,\cdot)\) \(\chi_{629}(523,\cdot)\) \(\chi_{629}(540,\cdot)\) \(\chi_{629}(574,\cdot)\) \(\chi_{629}(616,\cdot)\)
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: | 36.0.12858230878574949858207932230482249727234181545281621536549998606426152894209663387757389.1 |
Values on generators
\((445,409)\) → \((i,e\left(\frac{11}{36}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 629 }(13, a) \) | \(-1\) | \(1\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(1\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) |
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)