from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(134, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([64]))
pari: [g,chi] = znchar(Mod(17,134))
Basic properties
Modulus: | \(134\) | |
Conductor: | \(67\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(33\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{67}(17,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 134.g
\(\chi_{134}(17,\cdot)\) \(\chi_{134}(19,\cdot)\) \(\chi_{134}(21,\cdot)\) \(\chi_{134}(23,\cdot)\) \(\chi_{134}(33,\cdot)\) \(\chi_{134}(35,\cdot)\) \(\chi_{134}(39,\cdot)\) \(\chi_{134}(47,\cdot)\) \(\chi_{134}(49,\cdot)\) \(\chi_{134}(55,\cdot)\) \(\chi_{134}(65,\cdot)\) \(\chi_{134}(71,\cdot)\) \(\chi_{134}(73,\cdot)\) \(\chi_{134}(77,\cdot)\) \(\chi_{134}(83,\cdot)\) \(\chi_{134}(93,\cdot)\) \(\chi_{134}(103,\cdot)\) \(\chi_{134}(121,\cdot)\) \(\chi_{134}(123,\cdot)\) \(\chi_{134}(127,\cdot)\)
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 33 polynomial |
Values on generators
\(69\) → \(e\left(\frac{32}{33}\right)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 134 }(17, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{4}{33}\right)\) |
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)