from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6864, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,15,30,48,35]))
pari: [g,chi] = znchar(Mod(245,6864))
Basic properties
Modulus: | \(6864\) | |
Conductor: | \(6864\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | 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.kt
\(\chi_{6864}(245,\cdot)\) \(\chi_{6864}(509,\cdot)\) \(\chi_{6864}(917,\cdot)\) \(\chi_{6864}(1709,\cdot)\) \(\chi_{6864}(2117,\cdot)\) \(\chi_{6864}(2165,\cdot)\) \(\chi_{6864}(2381,\cdot)\) \(\chi_{6864}(2957,\cdot)\) \(\chi_{6864}(3413,\cdot)\) \(\chi_{6864}(4205,\cdot)\) \(\chi_{6864}(4613,\cdot)\) \(\chi_{6864}(4877,\cdot)\) \(\chi_{6864}(5285,\cdot)\) \(\chi_{6864}(5861,\cdot)\) \(\chi_{6864}(6077,\cdot)\) \(\chi_{6864}(6125,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((2575,1717,4577,4369,2641)\) → \((1,i,-1,e\left(\frac{4}{5}\right),e\left(\frac{7}{12}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) | \(37\) |
\( \chi_{ 6864 }(245, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{14}{15}\right)\) |
sage: chi.jacobi_sum(n)