from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5265, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([4,0,12]))
pari: [g,chi] = znchar(Mod(451,5265))
Basic properties
Modulus: | \(5265\) | |
Conductor: | \(351\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(9\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{351}(178,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5265.cb
\(\chi_{5265}(451,\cdot)\) \(\chi_{5265}(1576,\cdot)\) \(\chi_{5265}(2206,\cdot)\) \(\chi_{5265}(3331,\cdot)\) \(\chi_{5265}(3961,\cdot)\) \(\chi_{5265}(5086,\cdot)\)
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: | \(\Q(\zeta_{9})\) |
Fixed field: | 9.9.151470380950257681.2 |
Values on generators
\((326,2107,2836)\) → \((e\left(\frac{2}{9}\right),1,e\left(\frac{2}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
\( \chi_{ 5265 }(451, a) \) | \(1\) | \(1\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(1\) | \(e\left(\frac{4}{9}\right)\) |
sage: chi.jacobi_sum(n)