from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(221760, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,15,50,45,10,24]))
pari: [g,chi] = znchar(Mod(188303,221760))
Basic properties
| Modulus: | \(221760\) | |
| Conductor: | \(55440\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{55440}(8123,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 221760.cxs
\(\chi_{221760}(47,\cdot)\) \(\chi_{221760}(22223,\cdot)\) \(\chi_{221760}(25007,\cdot)\) \(\chi_{221760}(47183,\cdot)\) \(\chi_{221760}(60527,\cdot)\) \(\chi_{221760}(65327,\cdot)\) \(\chi_{221760}(82703,\cdot)\) \(\chi_{221760}(87503,\cdot)\) \(\chi_{221760}(105647,\cdot)\) \(\chi_{221760}(127823,\cdot)\) \(\chi_{221760}(141167,\cdot)\) \(\chi_{221760}(163343,\cdot)\) \(\chi_{221760}(166127,\cdot)\) \(\chi_{221760}(181487,\cdot)\) \(\chi_{221760}(188303,\cdot)\) \(\chi_{221760}(203663,\cdot)\)
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
\((48511,124741,98561,133057,190081,141121)\) → \((-1,i,e\left(\frac{5}{6}\right),-i,e\left(\frac{1}{6}\right),e\left(\frac{2}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 221760 }(188303, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(-i\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{60}\right)\) |
sage: chi.jacobi_sum(n)