from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6008, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([25,25,29]))
pari: [g,chi] = znchar(Mod(5955,6008))
Basic properties
| Modulus: | \(6008\) | |
| Conductor: | \(6008\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(50\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6008.bn
\(\chi_{6008}(195,\cdot)\) \(\chi_{6008}(331,\cdot)\) \(\chi_{6008}(403,\cdot)\) \(\chi_{6008}(1003,\cdot)\) \(\chi_{6008}(1027,\cdot)\) \(\chi_{6008}(1323,\cdot)\) \(\chi_{6008}(1331,\cdot)\) \(\chi_{6008}(1451,\cdot)\) \(\chi_{6008}(1587,\cdot)\) \(\chi_{6008}(1787,\cdot)\) \(\chi_{6008}(1803,\cdot)\) \(\chi_{6008}(2091,\cdot)\) \(\chi_{6008}(2523,\cdot)\) \(\chi_{6008}(2811,\cdot)\) \(\chi_{6008}(3803,\cdot)\) \(\chi_{6008}(4547,\cdot)\) \(\chi_{6008}(5523,\cdot)\) \(\chi_{6008}(5683,\cdot)\) \(\chi_{6008}(5891,\cdot)\) \(\chi_{6008}(5955,\cdot)\)
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
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 6008 }(5955, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{33}{50}\right)\) |
sage: chi.jacobi_sum(n)