from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(966, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,22,18]))
pari: [g,chi] = znchar(Mod(583,966))
Basic properties
Modulus: | \(966\) | |
Conductor: | \(161\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(33\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{161}(100,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 966.y
\(\chi_{966}(25,\cdot)\) \(\chi_{966}(121,\cdot)\) \(\chi_{966}(151,\cdot)\) \(\chi_{966}(163,\cdot)\) \(\chi_{966}(193,\cdot)\) \(\chi_{966}(289,\cdot)\) \(\chi_{966}(331,\cdot)\) \(\chi_{966}(361,\cdot)\) \(\chi_{966}(403,\cdot)\) \(\chi_{966}(445,\cdot)\) \(\chi_{966}(487,\cdot)\) \(\chi_{966}(499,\cdot)\) \(\chi_{966}(541,\cdot)\) \(\chi_{966}(583,\cdot)\) \(\chi_{966}(625,\cdot)\) \(\chi_{966}(739,\cdot)\) \(\chi_{966}(823,\cdot)\) \(\chi_{966}(877,\cdot)\) \(\chi_{966}(949,\cdot)\) \(\chi_{966}(961,\cdot)\)
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: | 33.33.277966181338944111003326058293667039541136678070715028736001.1 |
Values on generators
\((323,829,925)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{3}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 966 }(583, a) \) | \(1\) | \(1\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{3}{11}\right)\) |
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)