from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8036, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,19,0]))
pari: [g,chi] = znchar(Mod(1067,8036))
Basic properties
| Modulus: | \(8036\) | |
| Conductor: | \(196\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{196}(87,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8036.cx
\(\chi_{8036}(1067,\cdot)\) \(\chi_{8036}(1559,\cdot)\) \(\chi_{8036}(2215,\cdot)\) \(\chi_{8036}(2707,\cdot)\) \(\chi_{8036}(3855,\cdot)\) \(\chi_{8036}(4511,\cdot)\) \(\chi_{8036}(5003,\cdot)\) \(\chi_{8036}(5659,\cdot)\) \(\chi_{8036}(6151,\cdot)\) \(\chi_{8036}(6807,\cdot)\) \(\chi_{8036}(7299,\cdot)\) \(\chi_{8036}(7955,\cdot)\)
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: | \(\Q(\zeta_{196})^+\) |
Values on generators
\((4019,493,785)\) → \((-1,e\left(\frac{19}{42}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 8036 }(1067, a) \) | \(1\) | \(1\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{5}{21}\right)\) |
sage: chi.jacobi_sum(n)