from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(335, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([33,14]))
pari: [g,chi] = znchar(Mod(253,335))
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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 335.s
\(\chi_{335}(3,\cdot)\) \(\chi_{335}(8,\cdot)\) \(\chi_{335}(27,\cdot)\) \(\chi_{335}(42,\cdot)\) \(\chi_{335}(43,\cdot)\) \(\chi_{335}(52,\cdot)\) \(\chi_{335}(53,\cdot)\) \(\chi_{335}(58,\cdot)\) \(\chi_{335}(72,\cdot)\) \(\chi_{335}(112,\cdot)\) \(\chi_{335}(137,\cdot)\) \(\chi_{335}(142,\cdot)\) \(\chi_{335}(177,\cdot)\) \(\chi_{335}(187,\cdot)\) \(\chi_{335}(192,\cdot)\) \(\chi_{335}(228,\cdot)\) \(\chi_{335}(243,\cdot)\) \(\chi_{335}(253,\cdot)\) \(\chi_{335}(273,\cdot)\) \(\chi_{335}(313,\cdot)\)
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: | 44.44.5769681722973112639196821168549004411214468274227153473067543998508362785597448237240314483642578125.1 |
Values on generators
\((202,136)\) → \((-i,e\left(\frac{7}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
\( \chi_{ 335 }(253, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{13}{44}\right)\) |
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)