from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6003, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([35,21,36]))
pari: [g,chi] = znchar(Mod(3587,6003))
Basic properties
| Modulus: | \(6003\) | |
| Conductor: | \(6003\) | 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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6003.bw
\(\chi_{6003}(344,\cdot)\) \(\chi_{6003}(1379,\cdot)\) \(\chi_{6003}(1586,\cdot)\) \(\chi_{6003}(1793,\cdot)\) \(\chi_{6003}(2345,\cdot)\) \(\chi_{6003}(2414,\cdot)\) \(\chi_{6003}(3242,\cdot)\) \(\chi_{6003}(3380,\cdot)\) \(\chi_{6003}(3587,\cdot)\) \(\chi_{6003}(3794,\cdot)\) \(\chi_{6003}(4415,\cdot)\) \(\chi_{6003}(5243,\cdot)\)
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
\((668,3133,4555)\) → \((e\left(\frac{5}{6}\right),-1,e\left(\frac{6}{7}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 6003 }(3587, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) |
sage: chi.jacobi_sum(n)