from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(966, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,22,63]))
pari: [g,chi] = znchar(Mod(37,966))
Basic properties
| Modulus: | \(966\) | |
| Conductor: | \(161\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{161}(37,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 966.z
\(\chi_{966}(37,\cdot)\) \(\chi_{966}(67,\cdot)\) \(\chi_{966}(79,\cdot)\) \(\chi_{966}(109,\cdot)\) \(\chi_{966}(205,\cdot)\) \(\chi_{966}(235,\cdot)\) \(\chi_{966}(247,\cdot)\) \(\chi_{966}(319,\cdot)\) \(\chi_{966}(373,\cdot)\) \(\chi_{966}(457,\cdot)\) \(\chi_{966}(571,\cdot)\) \(\chi_{966}(613,\cdot)\) \(\chi_{966}(655,\cdot)\) \(\chi_{966}(697,\cdot)\) \(\chi_{966}(709,\cdot)\) \(\chi_{966}(751,\cdot)\) \(\chi_{966}(793,\cdot)\) \(\chi_{966}(835,\cdot)\) \(\chi_{966}(865,\cdot)\) \(\chi_{966}(907,\cdot)\)
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
\((323,829,925)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{21}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 966 }(37, a) \) | \(-1\) | \(1\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{5}{11}\right)\) |
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)