from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4100, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,11,14]))
pari: [g,chi] = znchar(Mod(523,4100))
Basic properties
| Modulus: | \(4100\) | |
| Conductor: | \(4100\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4100.dz
\(\chi_{4100}(523,\cdot)\) \(\chi_{4100}(1583,\cdot)\) \(\chi_{4100}(2327,\cdot)\) \(\chi_{4100}(2567,\cdot)\) \(\chi_{4100}(2587,\cdot)\) \(\chi_{4100}(2647,\cdot)\) \(\chi_{4100}(3303,\cdot)\) \(\chi_{4100}(4063,\cdot)\)
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: | 20.20.327084139915563306751566374270629882812500000000000000000000.3 |
Values on generators
\((2051,1477,3901)\) → \((-1,e\left(\frac{11}{20}\right),e\left(\frac{7}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 4100 }(523, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(i\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(-i\) | \(e\left(\frac{11}{20}\right)\) |
sage: chi.jacobi_sum(n)