from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2842, base_ring=CyclotomicField(14))
M = H._module
chi = DirichletCharacter(H, M([2,8]))
pari: [g,chi] = znchar(Mod(141,2842))
Basic properties
| Modulus: | \(2842\) | |
| Conductor: | \(1421\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(7\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1421}(141,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2842.r
\(\chi_{2842}(141,\cdot)\) \(\chi_{2842}(169,\cdot)\) \(\chi_{2842}(645,\cdot)\) \(\chi_{2842}(1009,\cdot)\) \(\chi_{2842}(1093,\cdot)\) \(\chi_{2842}(2829,\cdot)\)
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: | \(\Q(\zeta_{7})\) |
| Fixed field: | 7.7.8233120419813614521.3 |
Values on generators
\((1277,785)\) → \((e\left(\frac{1}{7}\right),e\left(\frac{4}{7}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 2842 }(141, a) \) | \(1\) | \(1\) | \(1\) | \(e\left(\frac{5}{7}\right)\) | \(1\) | \(1\) | \(1\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) |
sage: chi.jacobi_sum(n)