from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6760, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([0,13,0,21]))
pari: [g,chi] = znchar(Mod(181,6760))
Basic properties
Modulus: | \(6760\) | |
Conductor: | \(1352\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(26\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1352}(181,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6760.do
\(\chi_{6760}(181,\cdot)\) \(\chi_{6760}(701,\cdot)\) \(\chi_{6760}(1221,\cdot)\) \(\chi_{6760}(1741,\cdot)\) \(\chi_{6760}(2261,\cdot)\) \(\chi_{6760}(2781,\cdot)\) \(\chi_{6760}(3301,\cdot)\) \(\chi_{6760}(3821,\cdot)\) \(\chi_{6760}(4341,\cdot)\) \(\chi_{6760}(4861,\cdot)\) \(\chi_{6760}(5381,\cdot)\) \(\chi_{6760}(5901,\cdot)\)
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: | 26.26.2105688347980841837008408245509158917686052431514719567252107034624.1 |
Values on generators
\((5071,3381,4057,5241)\) → \((1,-1,1,e\left(\frac{21}{26}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
\( \chi_{ 6760 }(181, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(1\) | \(e\left(\frac{1}{13}\right)\) | \(1\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{21}{26}\right)\) |
sage: chi.jacobi_sum(n)