from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5265, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([26,0,9]))
pari: [g,chi] = znchar(Mod(121,5265))
Basic properties
Modulus: | \(5265\) | |
Conductor: | \(1053\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1053}(121,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5265.gs
\(\chi_{5265}(121,\cdot)\) \(\chi_{5265}(556,\cdot)\) \(\chi_{5265}(706,\cdot)\) \(\chi_{5265}(1141,\cdot)\) \(\chi_{5265}(1291,\cdot)\) \(\chi_{5265}(1726,\cdot)\) \(\chi_{5265}(1876,\cdot)\) \(\chi_{5265}(2311,\cdot)\) \(\chi_{5265}(2461,\cdot)\) \(\chi_{5265}(2896,\cdot)\) \(\chi_{5265}(3046,\cdot)\) \(\chi_{5265}(3481,\cdot)\) \(\chi_{5265}(3631,\cdot)\) \(\chi_{5265}(4066,\cdot)\) \(\chi_{5265}(4216,\cdot)\) \(\chi_{5265}(4651,\cdot)\) \(\chi_{5265}(4801,\cdot)\) \(\chi_{5265}(5236,\cdot)\)
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: | \(\Q(\zeta_{27})\) |
Fixed field: | Number field defined by a degree 54 polynomial |
Values on generators
\((326,2107,2836)\) → \((e\left(\frac{13}{27}\right),1,e\left(\frac{1}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
\( \chi_{ 5265 }(121, a) \) | \(1\) | \(1\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{29}{54}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{23}{54}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{2}{27}\right)\) |
sage: chi.jacobi_sum(n)