from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8112, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([13,13,13,23]))
pari: [g,chi] = znchar(Mod(311,8112))
Basic properties
| Modulus: | \(8112\) | |
| Conductor: | \(4056\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(26\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{4056}(2339,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8112.dl
\(\chi_{8112}(311,\cdot)\) \(\chi_{8112}(935,\cdot)\) \(\chi_{8112}(1559,\cdot)\) \(\chi_{8112}(2183,\cdot)\) \(\chi_{8112}(2807,\cdot)\) \(\chi_{8112}(3431,\cdot)\) \(\chi_{8112}(4679,\cdot)\) \(\chi_{8112}(5303,\cdot)\) \(\chi_{8112}(5927,\cdot)\) \(\chi_{8112}(6551,\cdot)\) \(\chi_{8112}(7175,\cdot)\) \(\chi_{8112}(7799,\cdot)\)
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: | 26.26.3357147364017859700104756459204898773121980170769842244620081043762839552.1 |
Values on generators
\((5071,6085,2705,3889)\) → \((-1,-1,-1,e\left(\frac{23}{26}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 8112 }(311, a) \) | \(1\) | \(1\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(-1\) | \(1\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{3}{26}\right)\) |
sage: chi.jacobi_sum(n)