from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2112, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([0,5,0,8]))
pari: [g,chi] = znchar(Mod(49,2112))
Basic properties
| Modulus: | \(2112\) | |
| 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 \(\chi_{176}(5,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2112.ci
\(\chi_{2112}(49,\cdot)\) \(\chi_{2112}(433,\cdot)\) \(\chi_{2112}(625,\cdot)\) \(\chi_{2112}(817,\cdot)\) \(\chi_{2112}(1105,\cdot)\) \(\chi_{2112}(1489,\cdot)\) \(\chi_{2112}(1681,\cdot)\) \(\chi_{2112}(1873,\cdot)\)
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: | 20.20.1655513490330868290261743826894848.1 |
Values on generators
\((2047,133,1409,1729)\) → \((1,i,1,e\left(\frac{2}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 2112 }(49, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(-1\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{3}{20}\right)\) |
sage: chi.jacobi_sum(n)