from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1287, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,48,5]))
pari: [g,chi] = znchar(Mod(1081,1287))
Basic properties
Modulus: | \(1287\) | |
Conductor: | \(143\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{143}(80,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1287.eg
\(\chi_{1287}(37,\cdot)\) \(\chi_{1287}(136,\cdot)\) \(\chi_{1287}(163,\cdot)\) \(\chi_{1287}(262,\cdot)\) \(\chi_{1287}(280,\cdot)\) \(\chi_{1287}(379,\cdot)\) \(\chi_{1287}(388,\cdot)\) \(\chi_{1287}(487,\cdot)\) \(\chi_{1287}(631,\cdot)\) \(\chi_{1287}(730,\cdot)\) \(\chi_{1287}(856,\cdot)\) \(\chi_{1287}(955,\cdot)\) \(\chi_{1287}(973,\cdot)\) \(\chi_{1287}(982,\cdot)\) \(\chi_{1287}(1072,\cdot)\) \(\chi_{1287}(1081,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((1145,937,496)\) → \((1,e\left(\frac{4}{5}\right),e\left(\frac{1}{12}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(14\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 1287 }(1081, a) \) | \(-1\) | \(1\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{49}{60}\right)\) |
sage: chi.jacobi_sum(n)