from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5400, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,18,8,27]))
pari: [g,chi] = znchar(Mod(43,5400))
Basic properties
| Modulus: | \(5400\) | |
| Conductor: | \(1080\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1080}(43,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5400.eh
\(\chi_{5400}(43,\cdot)\) \(\chi_{5400}(643,\cdot)\) \(\chi_{5400}(907,\cdot)\) \(\chi_{5400}(1507,\cdot)\) \(\chi_{5400}(1843,\cdot)\) \(\chi_{5400}(2443,\cdot)\) \(\chi_{5400}(2707,\cdot)\) \(\chi_{5400}(3307,\cdot)\) \(\chi_{5400}(3643,\cdot)\) \(\chi_{5400}(4243,\cdot)\) \(\chi_{5400}(4507,\cdot)\) \(\chi_{5400}(5107,\cdot)\)
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
\((1351,2701,1001,2377)\) → \((-1,-1,e\left(\frac{2}{9}\right),-i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 5400 }(43, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{7}{9}\right)\) |
sage: chi.jacobi_sum(n)