from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(335, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([33,32]))
pari: [g,chi] = znchar(Mod(263,335))
Basic properties
| Modulus: | \(335\) | |
| Conductor: | \(335\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | 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 335.r
\(\chi_{335}(22,\cdot)\) \(\chi_{335}(62,\cdot)\) \(\chi_{335}(82,\cdot)\) \(\chi_{335}(92,\cdot)\) \(\chi_{335}(107,\cdot)\) \(\chi_{335}(143,\cdot)\) \(\chi_{335}(148,\cdot)\) \(\chi_{335}(158,\cdot)\) \(\chi_{335}(193,\cdot)\) \(\chi_{335}(198,\cdot)\) \(\chi_{335}(223,\cdot)\) \(\chi_{335}(263,\cdot)\) \(\chi_{335}(277,\cdot)\) \(\chi_{335}(282,\cdot)\) \(\chi_{335}(283,\cdot)\) \(\chi_{335}(292,\cdot)\) \(\chi_{335}(293,\cdot)\) \(\chi_{335}(308,\cdot)\) \(\chi_{335}(327,\cdot)\) \(\chi_{335}(332,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
| Fixed field: | 44.0.1285293322114749975316734499565383027670855039925852856553251057809838000801391899585723876953125.1 |
Values on generators
\((202,136)\) → \((-i,e\left(\frac{8}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 335 }(263, a) \) | \(-1\) | \(1\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{3}{44}\right)\) |
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)