from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5265, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([4,9,9]))
pari: [g,chi] = znchar(Mod(307,5265))
Basic properties
Modulus: | \(5265\) | |
Conductor: | \(1755\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1755}(112,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5265.gg
\(\chi_{5265}(307,\cdot)\) \(\chi_{5265}(343,\cdot)\) \(\chi_{5265}(928,\cdot)\) \(\chi_{5265}(1477,\cdot)\) \(\chi_{5265}(2062,\cdot)\) \(\chi_{5265}(2098,\cdot)\) \(\chi_{5265}(2683,\cdot)\) \(\chi_{5265}(3232,\cdot)\) \(\chi_{5265}(3817,\cdot)\) \(\chi_{5265}(3853,\cdot)\) \(\chi_{5265}(4438,\cdot)\) \(\chi_{5265}(4987,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
Fixed field: | Number field defined by a degree 36 polynomial |
Values on generators
\((326,2107,2836)\) → \((e\left(\frac{1}{9}\right),i,i)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
\( \chi_{ 5265 }(307, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{29}{36}\right)\) |
sage: chi.jacobi_sum(n)