from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6003, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([0,0,1]))
pari: [g,chi] = znchar(Mod(4555,6003))
Basic properties
Modulus: | \(6003\) | |
Conductor: | \(29\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{29}(2,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6003.br
\(\chi_{6003}(1036,\cdot)\) \(\chi_{6003}(1864,\cdot)\) \(\chi_{6003}(3106,\cdot)\) \(\chi_{6003}(3520,\cdot)\) \(\chi_{6003}(3727,\cdot)\) \(\chi_{6003}(3934,\cdot)\) \(\chi_{6003}(4348,\cdot)\) \(\chi_{6003}(4555,\cdot)\) \(\chi_{6003}(4969,\cdot)\) \(\chi_{6003}(5176,\cdot)\) \(\chi_{6003}(5383,\cdot)\) \(\chi_{6003}(5797,\cdot)\)
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: | \(\Q(\zeta_{28})\) |
Fixed field: | Number field defined by a degree 28 polynomial |
Values on generators
\((668,3133,4555)\) → \((1,1,e\left(\frac{1}{28}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 6003 }(4555, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{1}{7}\right)\) |
sage: chi.jacobi_sum(n)