from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(966, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,11,12]))
pari: [g,chi] = znchar(Mod(73,966))
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 \(\chi_{161}(73,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 966.bb
\(\chi_{966}(31,\cdot)\) \(\chi_{966}(73,\cdot)\) \(\chi_{966}(187,\cdot)\) \(\chi_{966}(271,\cdot)\) \(\chi_{966}(325,\cdot)\) \(\chi_{966}(397,\cdot)\) \(\chi_{966}(409,\cdot)\) \(\chi_{966}(439,\cdot)\) \(\chi_{966}(535,\cdot)\) \(\chi_{966}(565,\cdot)\) \(\chi_{966}(577,\cdot)\) \(\chi_{966}(607,\cdot)\) \(\chi_{966}(703,\cdot)\) \(\chi_{966}(745,\cdot)\) \(\chi_{966}(775,\cdot)\) \(\chi_{966}(817,\cdot)\) \(\chi_{966}(859,\cdot)\) \(\chi_{966}(901,\cdot)\) \(\chi_{966}(913,\cdot)\) \(\chi_{966}(955,\cdot)\)
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
\((323,829,925)\) → \((1,e\left(\frac{1}{6}\right),e\left(\frac{2}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 966 }(73, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{15}{22}\right)\) |
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)