from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5635, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,41,21]))
pari: [g,chi] = znchar(Mod(229,5635))
Basic properties
| Modulus: | \(5635\) | |
| Conductor: | \(5635\) | 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 5635.bw
\(\chi_{5635}(229,\cdot)\) \(\chi_{5635}(689,\cdot)\) \(\chi_{5635}(1034,\cdot)\) \(\chi_{5635}(1494,\cdot)\) \(\chi_{5635}(1839,\cdot)\) \(\chi_{5635}(2299,\cdot)\) \(\chi_{5635}(2644,\cdot)\) \(\chi_{5635}(3104,\cdot)\) \(\chi_{5635}(3909,\cdot)\) \(\chi_{5635}(4254,\cdot)\) \(\chi_{5635}(4714,\cdot)\) \(\chi_{5635}(5059,\cdot)\)
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
\((3382,346,2696)\) → \((-1,e\left(\frac{41}{42}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 5635 }(229, a) \) | \(1\) | \(1\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{11}{21}\right)\) |
sage: chi.jacobi_sum(n)