from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5082, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,55,51]))
pari: [g,chi] = znchar(Mod(439,5082))
Basic properties
Modulus: | \(5082\) | |
Conductor: | \(847\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{847}(439,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5082.br
\(\chi_{5082}(439,\cdot)\) \(\chi_{5082}(703,\cdot)\) \(\chi_{5082}(901,\cdot)\) \(\chi_{5082}(1165,\cdot)\) \(\chi_{5082}(1363,\cdot)\) \(\chi_{5082}(1627,\cdot)\) \(\chi_{5082}(1825,\cdot)\) \(\chi_{5082}(2089,\cdot)\) \(\chi_{5082}(2287,\cdot)\) \(\chi_{5082}(2551,\cdot)\) \(\chi_{5082}(2749,\cdot)\) \(\chi_{5082}(3013,\cdot)\) \(\chi_{5082}(3211,\cdot)\) \(\chi_{5082}(3475,\cdot)\) \(\chi_{5082}(3673,\cdot)\) \(\chi_{5082}(3937,\cdot)\) \(\chi_{5082}(4135,\cdot)\) \(\chi_{5082}(4399,\cdot)\) \(\chi_{5082}(4861,\cdot)\) \(\chi_{5082}(5059,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
Fixed field: | Number field defined by a degree 66 polynomial |
Values on generators
\((3389,4357,2059)\) → \((1,e\left(\frac{5}{6}\right),e\left(\frac{17}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 5082 }(439, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{3}{11}\right)\) |
sage: chi.jacobi_sum(n)