from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7800, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,10,0,16,15]))
pari: [g,chi] = znchar(Mod(811,7800))
Basic properties
| Modulus: | \(7800\) | |
| Conductor: | \(2600\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2600}(811,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7800.im
\(\chi_{7800}(811,\cdot)\) \(\chi_{7800}(931,\cdot)\) \(\chi_{7800}(2371,\cdot)\) \(\chi_{7800}(2491,\cdot)\) \(\chi_{7800}(3931,\cdot)\) \(\chi_{7800}(5491,\cdot)\) \(\chi_{7800}(5611,\cdot)\) \(\chi_{7800}(7171,\cdot)\)
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: | Number field defined by a degree 20 polynomial |
Values on generators
\((1951,3901,5201,7177,4201)\) → \((-1,-1,1,e\left(\frac{4}{5}\right),-i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 7800 }(811, a) \) | \(1\) | \(1\) | \(-i\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)