from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4560, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,27,0,18,14]))
pari: [g,chi] = znchar(Mod(109,4560))
Basic properties
| Modulus: | \(4560\) | |
| 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 \(\chi_{1520}(109,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4560.hr
\(\chi_{4560}(109,\cdot)\) \(\chi_{4560}(469,\cdot)\) \(\chi_{4560}(1549,\cdot)\) \(\chi_{4560}(1789,\cdot)\) \(\chi_{4560}(2029,\cdot)\) \(\chi_{4560}(2149,\cdot)\) \(\chi_{4560}(2389,\cdot)\) \(\chi_{4560}(2749,\cdot)\) \(\chi_{4560}(3829,\cdot)\) \(\chi_{4560}(4069,\cdot)\) \(\chi_{4560}(4309,\cdot)\) \(\chi_{4560}(4429,\cdot)\)
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
\((1711,1141,3041,2737,1921)\) → \((1,-i,1,-1,e\left(\frac{7}{18}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 4560 }(109, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(-i\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{17}{36}\right)\) |
sage: chi.jacobi_sum(n)