from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6080, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,9,18,26]))
pari: [g,chi] = znchar(Mod(79,6080))
Basic properties
Modulus: | \(6080\) | |
Conductor: | \(1520\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1520}(459,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6080.gj
\(\chi_{6080}(79,\cdot)\) \(\chi_{6080}(1039,\cdot)\) \(\chi_{6080}(1199,\cdot)\) \(\chi_{6080}(1359,\cdot)\) \(\chi_{6080}(1839,\cdot)\) \(\chi_{6080}(2959,\cdot)\) \(\chi_{6080}(3119,\cdot)\) \(\chi_{6080}(4079,\cdot)\) \(\chi_{6080}(4239,\cdot)\) \(\chi_{6080}(4399,\cdot)\) \(\chi_{6080}(4879,\cdot)\) \(\chi_{6080}(5999,\cdot)\)
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
\((191,5701,1217,1921)\) → \((-1,i,-1,e\left(\frac{13}{18}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
\( \chi_{ 6080 }(79, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{36}\right)\) |
sage: chi.jacobi_sum(n)