from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6864, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,15,10,8,10]))
pari: [g,chi] = znchar(Mod(467,6864))
Basic properties
| Modulus: | \(6864\) | |
| Conductor: | \(6864\) | 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 6864.iv
\(\chi_{6864}(467,\cdot)\) \(\chi_{6864}(779,\cdot)\) \(\chi_{6864}(2027,\cdot)\) \(\chi_{6864}(2963,\cdot)\) \(\chi_{6864}(3899,\cdot)\) \(\chi_{6864}(4211,\cdot)\) \(\chi_{6864}(5459,\cdot)\) \(\chi_{6864}(6395,\cdot)\)
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
\((2575,1717,4577,4369,2641)\) → \((-1,-i,-1,e\left(\frac{2}{5}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) | \(37\) |
| \( \chi_{ 6864 }(467, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(-1\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) |
sage: chi.jacobi_sum(n)