from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4224, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,5,0,12]))
pari: [g,chi] = znchar(Mod(31,4224))
Basic properties
| Modulus: | \(4224\) | |
| Conductor: | \(176\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{176}(75,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4224.cf
\(\chi_{4224}(31,\cdot)\) \(\chi_{4224}(223,\cdot)\) \(\chi_{4224}(1567,\cdot)\) \(\chi_{4224}(1951,\cdot)\) \(\chi_{4224}(2143,\cdot)\) \(\chi_{4224}(2335,\cdot)\) \(\chi_{4224}(3679,\cdot)\) \(\chi_{4224}(4063,\cdot)\)
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: | 20.0.1655513490330868290261743826894848.1 |
Values on generators
\((2047,133,1409,3841)\) → \((-1,i,1,e\left(\frac{3}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 4224 }(31, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{17}{20}\right)\) |
sage: chi.jacobi_sum(n)