from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(629, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([9,32]))
pari: [g,chi] = znchar(Mod(10,629))
Basic properties
| Modulus: | \(629\) | |
| Conductor: | \(629\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | 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.br
\(\chi_{629}(10,\cdot)\) \(\chi_{629}(63,\cdot)\) \(\chi_{629}(158,\cdot)\) \(\chi_{629}(211,\cdot)\) \(\chi_{629}(232,\cdot)\) \(\chi_{629}(248,\cdot)\) \(\chi_{629}(269,\cdot)\) \(\chi_{629}(343,\cdot)\) \(\chi_{629}(380,\cdot)\) \(\chi_{629}(396,\cdot)\) \(\chi_{629}(454,\cdot)\) \(\chi_{629}(470,\cdot)\) \(\chi_{629}(507,\cdot)\) \(\chi_{629}(581,\cdot)\) \(\chi_{629}(602,\cdot)\) \(\chi_{629}(618,\cdot)\)
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: | \(\Q(\zeta_{48})\) |
| Fixed field: | Number field defined by a degree 48 polynomial |
Values on generators
\((445,409)\) → \((e\left(\frac{3}{16}\right),e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 629 }(10, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) |
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)