from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6017, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,41]))
pari: [g,chi] = znchar(Mod(5512,6017))
Basic properties
| Modulus: | \(6017\) | |
| Conductor: | \(547\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{547}(42,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6017.bc
\(\chi_{6017}(430,\cdot)\) \(\chi_{6017}(925,\cdot)\) \(\chi_{6017}(1761,\cdot)\) \(\chi_{6017}(1992,\cdot)\) \(\chi_{6017}(3268,\cdot)\) \(\chi_{6017}(4016,\cdot)\) \(\chi_{6017}(4797,\cdot)\) \(\chi_{6017}(4962,\cdot)\) \(\chi_{6017}(5292,\cdot)\) \(\chi_{6017}(5347,\cdot)\) \(\chi_{6017}(5457,\cdot)\) \(\chi_{6017}(5512,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
| Fixed field: | Number field defined by a degree 42 polynomial |
Values on generators
\((3830,2190)\) → \((1,e\left(\frac{41}{42}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 6017 }(5512, a) \) | \(-1\) | \(1\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{31}{42}\right)\) |
sage: chi.jacobi_sum(n)