from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6042, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([13,0,8]))
pari: [g,chi] = znchar(Mod(3155,6042))
Basic properties
Modulus: | \(6042\) | |
Conductor: | \(159\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(26\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{159}(134,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6042.bm
\(\chi_{6042}(77,\cdot)\) \(\chi_{6042}(1901,\cdot)\) \(\chi_{6042}(2699,\cdot)\) \(\chi_{6042}(3155,\cdot)\) \(\chi_{6042}(3269,\cdot)\) \(\chi_{6042}(3383,\cdot)\) \(\chi_{6042}(3725,\cdot)\) \(\chi_{6042}(4181,\cdot)\) \(\chi_{6042}(4409,\cdot)\) \(\chi_{6042}(4865,\cdot)\) \(\chi_{6042}(5207,\cdot)\) \(\chi_{6042}(6005,\cdot)\)
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: | 26.0.384766437057818380952237905666104641217782272563.1 |
Values on generators
\((2015,4771,2281)\) → \((-1,1,e\left(\frac{4}{13}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
\( \chi_{ 6042 }(3155, a) \) | \(-1\) | \(1\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(-1\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{7}{26}\right)\) |
sage: chi.jacobi_sum(n)