from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6017, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,20]))
pari: [g,chi] = znchar(Mod(505,6017))
Basic properties
| Modulus: | \(6017\) | |
| Conductor: | \(6017\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6017.bb
\(\chi_{6017}(505,\cdot)\) \(\chi_{6017}(560,\cdot)\) \(\chi_{6017}(670,\cdot)\) \(\chi_{6017}(725,\cdot)\) \(\chi_{6017}(1055,\cdot)\) \(\chi_{6017}(1220,\cdot)\) \(\chi_{6017}(2001,\cdot)\) \(\chi_{6017}(2749,\cdot)\) \(\chi_{6017}(4025,\cdot)\) \(\chi_{6017}(4256,\cdot)\) \(\chi_{6017}(5092,\cdot)\) \(\chi_{6017}(5587,\cdot)\)
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
\((3830,2190)\) → \((-1,e\left(\frac{10}{21}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 6017 }(505, a) \) | \(-1\) | \(1\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{5}{21}\right)\) |
sage: chi.jacobi_sum(n)