from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6018, base_ring=CyclotomicField(8))
M = H._module
chi = DirichletCharacter(H, M([4,3,4]))
pari: [g,chi] = znchar(Mod(2123,6018))
Basic properties
Modulus: | \(6018\) | |
Conductor: | \(3009\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(8\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{3009}(2123,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6018.o
\(\chi_{6018}(2123,\cdot)\) \(\chi_{6018}(2831,\cdot)\) \(\chi_{6018}(4955,\cdot)\) \(\chi_{6018}(5663,\cdot)\)
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: | \(\Q(\zeta_{8})\) |
Fixed field: | Number field defined by a degree 8 polynomial |
Values on generators
\((4013,1771,1123)\) → \((-1,e\left(\frac{3}{8}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
\( \chi_{ 6018 }(2123, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(1\) | \(i\) | \(e\left(\frac{5}{8}\right)\) | \(-i\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)