from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8085, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,0,26,21]))
pari: [g,chi] = znchar(Mod(296,8085))
Basic properties
| Modulus: | \(8085\) | |
| Conductor: | \(1617\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1617}(296,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8085.ey
\(\chi_{8085}(296,\cdot)\) \(\chi_{8085}(956,\cdot)\) \(\chi_{8085}(2111,\cdot)\) \(\chi_{8085}(2606,\cdot)\) \(\chi_{8085}(3266,\cdot)\) \(\chi_{8085}(3761,\cdot)\) \(\chi_{8085}(4421,\cdot)\) \(\chi_{8085}(4916,\cdot)\) \(\chi_{8085}(5576,\cdot)\) \(\chi_{8085}(6071,\cdot)\) \(\chi_{8085}(7226,\cdot)\) \(\chi_{8085}(7886,\cdot)\)
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
\((2696,4852,1816,3676)\) → \((-1,1,e\left(\frac{13}{21}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(13\) | \(16\) | \(17\) | \(19\) | \(23\) | \(26\) | \(29\) |
| \( \chi_{ 8085 }(296, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{1}{7}\right)\) |
sage: chi.jacobi_sum(n)