from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8112, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([13,0,13,5]))
pari: [g,chi] = znchar(Mod(6239,8112))
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 \(\chi_{2028}(155,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8112.df
\(\chi_{8112}(623,\cdot)\) \(\chi_{8112}(1247,\cdot)\) \(\chi_{8112}(1871,\cdot)\) \(\chi_{8112}(2495,\cdot)\) \(\chi_{8112}(3119,\cdot)\) \(\chi_{8112}(3743,\cdot)\) \(\chi_{8112}(4367,\cdot)\) \(\chi_{8112}(4991,\cdot)\) \(\chi_{8112}(5615,\cdot)\) \(\chi_{8112}(6239,\cdot)\) \(\chi_{8112}(6863,\cdot)\) \(\chi_{8112}(7487,\cdot)\)
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: | 26.26.409808027834211389172943903711535494765866720064678008376474736787456.1 |
Values on generators
\((5071,6085,2705,3889)\) → \((-1,1,-1,e\left(\frac{5}{26}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 8112 }(6239, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(1\) | \(1\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) |
sage: chi.jacobi_sum(n)