from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6864, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,0,15,27,5]))
pari: [g,chi] = znchar(Mod(17,6864))
Basic properties
| Modulus: | \(6864\) | |
| Conductor: | \(429\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{429}(17,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6864.jv
\(\chi_{6864}(17,\cdot)\) \(\chi_{6864}(1889,\cdot)\) \(\chi_{6864}(2129,\cdot)\) \(\chi_{6864}(3137,\cdot)\) \(\chi_{6864}(4001,\cdot)\) \(\chi_{6864}(4385,\cdot)\) \(\chi_{6864}(5249,\cdot)\) \(\chi_{6864}(6497,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
| Fixed field: | Number field defined by a degree 30 polynomial |
Values on generators
\((2575,1717,4577,4369,2641)\) → \((1,1,-1,e\left(\frac{9}{10}\right),e\left(\frac{1}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) | \(37\) |
| \( \chi_{ 6864 }(17, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{29}{30}\right)\) |
sage: chi.jacobi_sum(n)