from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5185, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,0,1]))
pari: [g,chi] = znchar(Mod(2381,5185))
Basic properties
| Modulus: | \(5185\) | |
| Conductor: | \(61\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{61}(2,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5185.ih
\(\chi_{5185}(681,\cdot)\) \(\chi_{5185}(1446,\cdot)\) \(\chi_{5185}(1531,\cdot)\) \(\chi_{5185}(1616,\cdot)\) \(\chi_{5185}(1701,\cdot)\) \(\chi_{5185}(1786,\cdot)\) \(\chi_{5185}(2381,\cdot)\) \(\chi_{5185}(2466,\cdot)\) \(\chi_{5185}(3146,\cdot)\) \(\chi_{5185}(3231,\cdot)\) \(\chi_{5185}(3826,\cdot)\) \(\chi_{5185}(3911,\cdot)\) \(\chi_{5185}(3996,\cdot)\) \(\chi_{5185}(4081,\cdot)\) \(\chi_{5185}(4166,\cdot)\) \(\chi_{5185}(4931,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((3112,4576,2381)\) → \((1,1,e\left(\frac{1}{60}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 5185 }(2381, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(i\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{2}{3}\right)\) |
sage: chi.jacobi_sum(n)