from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5265, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([4,0,27]))
pari: [g,chi] = znchar(Mod(4471,5265))
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 \(\chi_{1053}(259,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5265.gu
\(\chi_{5265}(376,\cdot)\) \(\chi_{5265}(571,\cdot)\) \(\chi_{5265}(961,\cdot)\) \(\chi_{5265}(1156,\cdot)\) \(\chi_{5265}(1546,\cdot)\) \(\chi_{5265}(1741,\cdot)\) \(\chi_{5265}(2131,\cdot)\) \(\chi_{5265}(2326,\cdot)\) \(\chi_{5265}(2716,\cdot)\) \(\chi_{5265}(2911,\cdot)\) \(\chi_{5265}(3301,\cdot)\) \(\chi_{5265}(3496,\cdot)\) \(\chi_{5265}(3886,\cdot)\) \(\chi_{5265}(4081,\cdot)\) \(\chi_{5265}(4471,\cdot)\) \(\chi_{5265}(4666,\cdot)\) \(\chi_{5265}(5056,\cdot)\) \(\chi_{5265}(5251,\cdot)\)
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
\((326,2107,2836)\) → \((e\left(\frac{2}{27}\right),1,-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
| \( \chi_{ 5265 }(4471, a) \) | \(1\) | \(1\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{25}{54}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{1}{27}\right)\) |
sage: chi.jacobi_sum(n)