from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6272, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([0,7,12]))
pari: [g,chi] = znchar(Mod(225,6272))
Basic properties
| Modulus: | \(6272\) | |
| Conductor: | \(784\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{784}(421,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6272.bt
\(\chi_{6272}(225,\cdot)\) \(\chi_{6272}(673,\cdot)\) \(\chi_{6272}(1121,\cdot)\) \(\chi_{6272}(2017,\cdot)\) \(\chi_{6272}(2465,\cdot)\) \(\chi_{6272}(2913,\cdot)\) \(\chi_{6272}(3361,\cdot)\) \(\chi_{6272}(3809,\cdot)\) \(\chi_{6272}(4257,\cdot)\) \(\chi_{6272}(5153,\cdot)\) \(\chi_{6272}(5601,\cdot)\) \(\chi_{6272}(6049,\cdot)\)
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
\((4607,3333,4609)\) → \((1,i,e\left(\frac{3}{7}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 6272 }(225, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(-i\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{5}{14}\right)\) |
sage: chi.jacobi_sum(n)