from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1288, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,0,55,21]))
pari: [g,chi] = znchar(Mod(201,1288))
Basic properties
Modulus: | \(1288\) | |
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 \(\chi_{161}(40,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1288.cc
\(\chi_{1288}(17,\cdot)\) \(\chi_{1288}(33,\cdot)\) \(\chi_{1288}(89,\cdot)\) \(\chi_{1288}(129,\cdot)\) \(\chi_{1288}(145,\cdot)\) \(\chi_{1288}(201,\cdot)\) \(\chi_{1288}(241,\cdot)\) \(\chi_{1288}(297,\cdot)\) \(\chi_{1288}(313,\cdot)\) \(\chi_{1288}(425,\cdot)\) \(\chi_{1288}(465,\cdot)\) \(\chi_{1288}(481,\cdot)\) \(\chi_{1288}(521,\cdot)\) \(\chi_{1288}(649,\cdot)\) \(\chi_{1288}(705,\cdot)\) \(\chi_{1288}(801,\cdot)\) \(\chi_{1288}(985,\cdot)\) \(\chi_{1288}(1137,\cdot)\) \(\chi_{1288}(1193,\cdot)\) \(\chi_{1288}(1249,\cdot)\)
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
\((967,645,185,281)\) → \((1,1,e\left(\frac{5}{6}\right),e\left(\frac{7}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(25\) | \(27\) |
\( \chi_{ 1288 }(201, a) \) | \(1\) | \(1\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{17}{22}\right)\) |
sage: chi.jacobi_sum(n)