from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1080, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,0,4,9]))
pari: [g,chi] = znchar(Mod(247,1080))
Basic properties
Modulus: | \(1080\) | |
Conductor: | \(540\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{540}(247,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1080.cp
\(\chi_{1080}(7,\cdot)\) \(\chi_{1080}(103,\cdot)\) \(\chi_{1080}(223,\cdot)\) \(\chi_{1080}(247,\cdot)\) \(\chi_{1080}(367,\cdot)\) \(\chi_{1080}(463,\cdot)\) \(\chi_{1080}(583,\cdot)\) \(\chi_{1080}(607,\cdot)\) \(\chi_{1080}(727,\cdot)\) \(\chi_{1080}(823,\cdot)\) \(\chi_{1080}(943,\cdot)\) \(\chi_{1080}(967,\cdot)\)
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
\((271,541,1001,217)\) → \((-1,1,e\left(\frac{1}{9}\right),i)\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 1080 }(247, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{8}{9}\right)\) |
sage: chi.jacobi_sum(n)