from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6003, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([35,21,39]))
pari: [g,chi] = znchar(Mod(689,6003))
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.ca
\(\chi_{6003}(689,\cdot)\) \(\chi_{6003}(758,\cdot)\) \(\chi_{6003}(1310,\cdot)\) \(\chi_{6003}(1517,\cdot)\) \(\chi_{6003}(1724,\cdot)\) \(\chi_{6003}(2759,\cdot)\) \(\chi_{6003}(3863,\cdot)\) \(\chi_{6003}(4691,\cdot)\) \(\chi_{6003}(5312,\cdot)\) \(\chi_{6003}(5519,\cdot)\) \(\chi_{6003}(5726,\cdot)\) \(\chi_{6003}(5864,\cdot)\)
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
\((668,3133,4555)\) → \((e\left(\frac{5}{6}\right),-1,e\left(\frac{13}{14}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 6003 }(689, a) \) | \(1\) | \(1\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{1}{21}\right)\) |
sage: chi.jacobi_sum(n)