from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(221760, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,0,0,5,0,18]))
pari: [g,chi] = znchar(Mod(127,221760))
Basic properties
| Modulus: | \(221760\) | |
| Conductor: | \(220\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{220}(127,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 221760.bok
\(\chi_{221760}(127,\cdot)\) \(\chi_{221760}(12223,\cdot)\) \(\chi_{221760}(52543,\cdot)\) \(\chi_{221760}(60607,\cdot)\) \(\chi_{221760}(100927,\cdot)\) \(\chi_{221760}(133183,\cdot)\) \(\chi_{221760}(141247,\cdot)\) \(\chi_{221760}(193663,\cdot)\)
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: | 20.0.177917354031751407392000000000000000.1 |
Values on generators
\((48511,124741,98561,133057,190081,141121)\) → \((-1,1,1,i,1,e\left(\frac{9}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 221760 }(127, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(i\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(-i\) | \(e\left(\frac{19}{20}\right)\) |
sage: chi.jacobi_sum(n)