from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6008, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([25,0,29]))
pari: [g,chi] = znchar(Mod(2951,6008))
Basic properties
Modulus: | \(6008\) | |
Conductor: | \(3004\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(50\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{3004}(2951,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6008.bi
\(\chi_{6008}(799,\cdot)\) \(\chi_{6008}(1543,\cdot)\) \(\chi_{6008}(2519,\cdot)\) \(\chi_{6008}(2679,\cdot)\) \(\chi_{6008}(2887,\cdot)\) \(\chi_{6008}(2951,\cdot)\) \(\chi_{6008}(3199,\cdot)\) \(\chi_{6008}(3335,\cdot)\) \(\chi_{6008}(3407,\cdot)\) \(\chi_{6008}(4007,\cdot)\) \(\chi_{6008}(4031,\cdot)\) \(\chi_{6008}(4327,\cdot)\) \(\chi_{6008}(4335,\cdot)\) \(\chi_{6008}(4455,\cdot)\) \(\chi_{6008}(4591,\cdot)\) \(\chi_{6008}(4791,\cdot)\) \(\chi_{6008}(4807,\cdot)\) \(\chi_{6008}(5095,\cdot)\) \(\chi_{6008}(5527,\cdot)\) \(\chi_{6008}(5815,\cdot)\)
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: | \(\Q(\zeta_{25})\) |
Fixed field: | Number field defined by a degree 50 polynomial |
Values on generators
\((1503,3005,1505)\) → \((-1,1,e\left(\frac{29}{50}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 6008 }(2951, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{4}{25}\right)\) |
sage: chi.jacobi_sum(n)