from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6864, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,0,0,6,5]))
pari: [g,chi] = znchar(Mod(145,6864))
Basic properties
| Modulus: | \(6864\) | |
| Conductor: | \(143\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{143}(2,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6864.lo
\(\chi_{6864}(145,\cdot)\) \(\chi_{6864}(193,\cdot)\) \(\chi_{6864}(721,\cdot)\) \(\chi_{6864}(865,\cdot)\) \(\chi_{6864}(1393,\cdot)\) \(\chi_{6864}(2065,\cdot)\) \(\chi_{6864}(2593,\cdot)\) \(\chi_{6864}(3313,\cdot)\) \(\chi_{6864}(3361,\cdot)\) \(\chi_{6864}(3841,\cdot)\) \(\chi_{6864}(3889,\cdot)\) \(\chi_{6864}(4561,\cdot)\) \(\chi_{6864}(5089,\cdot)\) \(\chi_{6864}(5233,\cdot)\) \(\chi_{6864}(5761,\cdot)\) \(\chi_{6864}(6481,\cdot)\)
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
\((2575,1717,4577,4369,2641)\) → \((1,1,1,e\left(\frac{1}{10}\right),e\left(\frac{1}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) | \(37\) |
| \( \chi_{ 6864 }(145, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{47}{60}\right)\) |
sage: chi.jacobi_sum(n)