from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5610, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,0,14,15]))
pari: [g,chi] = znchar(Mod(4781,5610))
Basic properties
| Modulus: | \(5610\) | |
| Conductor: | \(561\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{561}(293,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5610.eb
\(\chi_{5610}(701,\cdot)\) \(\chi_{5610}(761,\cdot)\) \(\chi_{5610}(1271,\cdot)\) \(\chi_{5610}(2741,\cdot)\) \(\chi_{5610}(2801,\cdot)\) \(\chi_{5610}(3251,\cdot)\) \(\chi_{5610}(4331,\cdot)\) \(\chi_{5610}(4781,\cdot)\)
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: | \(\Q(\zeta_{20})\) |
| Fixed field: | Number field defined by a degree 20 polynomial |
Values on generators
\((1871,3367,1531,3301)\) → \((-1,1,e\left(\frac{7}{10}\right),-i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(13\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 5610 }(4781, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(-i\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(1\) | \(e\left(\frac{1}{10}\right)\) |
sage: chi.jacobi_sum(n)