from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6048, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,27,28,12]))
pari: [g,chi] = znchar(Mod(1129,6048))
Basic properties
| Modulus: | \(6048\) | |
| Conductor: | \(3024\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{3024}(1885,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6048.iv
\(\chi_{6048}(25,\cdot)\) \(\chi_{6048}(121,\cdot)\) \(\chi_{6048}(1033,\cdot)\) \(\chi_{6048}(1129,\cdot)\) \(\chi_{6048}(2041,\cdot)\) \(\chi_{6048}(2137,\cdot)\) \(\chi_{6048}(3049,\cdot)\) \(\chi_{6048}(3145,\cdot)\) \(\chi_{6048}(4057,\cdot)\) \(\chi_{6048}(4153,\cdot)\) \(\chi_{6048}(5065,\cdot)\) \(\chi_{6048}(5161,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
| Fixed field: | Number field defined by a degree 36 polynomial |
Values on generators
\((4159,3781,3809,2593)\) → \((1,-i,e\left(\frac{7}{9}\right),e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 6048 }(1129, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(1\) | \(i\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{12}\right)\) |
sage: chi.jacobi_sum(n)