from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9200, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,5,14,10]))
pari: [g,chi] = znchar(Mod(459,9200))
Basic properties
| Modulus: | \(9200\) | |
| Conductor: | \(9200\) | 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 9200.cv
\(\chi_{9200}(459,\cdot)\) \(\chi_{9200}(1379,\cdot)\) \(\chi_{9200}(3219,\cdot)\) \(\chi_{9200}(4139,\cdot)\) \(\chi_{9200}(5059,\cdot)\) \(\chi_{9200}(5979,\cdot)\) \(\chi_{9200}(7819,\cdot)\) \(\chi_{9200}(8739,\cdot)\)
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
\((1151,6901,2577,1201)\) → \((-1,i,e\left(\frac{7}{10}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
| \( \chi_{ 9200 }(459, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) |
sage: chi.jacobi_sum(n)