from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8112, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([13,0,13,6]))
pari: [g,chi] = znchar(Mod(287,8112))
Basic properties
| Modulus: | \(8112\) | |
| Conductor: | \(2028\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(26\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2028}(287,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8112.dp
\(\chi_{8112}(287,\cdot)\) \(\chi_{8112}(911,\cdot)\) \(\chi_{8112}(1535,\cdot)\) \(\chi_{8112}(2159,\cdot)\) \(\chi_{8112}(2783,\cdot)\) \(\chi_{8112}(3407,\cdot)\) \(\chi_{8112}(4031,\cdot)\) \(\chi_{8112}(4655,\cdot)\) \(\chi_{8112}(5279,\cdot)\) \(\chi_{8112}(5903,\cdot)\) \(\chi_{8112}(6527,\cdot)\) \(\chi_{8112}(7151,\cdot)\)
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: | \(\Q(\zeta_{13})\) |
| Fixed field: | Number field defined by a degree 26 polynomial |
Values on generators
\((5071,6085,2705,3889)\) → \((-1,1,-1,e\left(\frac{3}{13}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 8112 }(287, a) \) | \(1\) | \(1\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{10}{13}\right)\) |
sage: chi.jacobi_sum(n)