from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(966, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,44,63]))
pari: [g,chi] = znchar(Mod(221,966))
Basic properties
Modulus: | \(966\) | |
Conductor: | \(483\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{483}(221,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 966.bf
\(\chi_{966}(11,\cdot)\) \(\chi_{966}(53,\cdot)\) \(\chi_{966}(65,\cdot)\) \(\chi_{966}(107,\cdot)\) \(\chi_{966}(149,\cdot)\) \(\chi_{966}(191,\cdot)\) \(\chi_{966}(221,\cdot)\) \(\chi_{966}(263,\cdot)\) \(\chi_{966}(359,\cdot)\) \(\chi_{966}(389,\cdot)\) \(\chi_{966}(401,\cdot)\) \(\chi_{966}(431,\cdot)\) \(\chi_{966}(527,\cdot)\) \(\chi_{966}(557,\cdot)\) \(\chi_{966}(569,\cdot)\) \(\chi_{966}(641,\cdot)\) \(\chi_{966}(695,\cdot)\) \(\chi_{966}(779,\cdot)\) \(\chi_{966}(893,\cdot)\) \(\chi_{966}(935,\cdot)\)
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: | Number field defined by a degree 66 polynomial |
Values on generators
\((323,829,925)\) → \((-1,e\left(\frac{2}{3}\right),e\left(\frac{21}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 966 }(221, a) \) | \(1\) | \(1\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{21}{22}\right)\) |
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)