from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4033, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([36,46]))
pari: [g,chi] = znchar(Mod(2637,4033))
Basic properties
| Modulus: | \(4033\) | |
| Conductor: | \(4033\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(27\) | 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 4033.dy
\(\chi_{4033}(26,\cdot)\) \(\chi_{4033}(158,\cdot)\) \(\chi_{4033}(676,\cdot)\) \(\chi_{4033}(766,\cdot)\) \(\chi_{4033}(877,\cdot)\) \(\chi_{4033}(988,\cdot)\) \(\chi_{4033}(1062,\cdot)\) \(\chi_{4033}(1247,\cdot)\) \(\chi_{4033}(1490,\cdot)\) \(\chi_{4033}(1506,\cdot)\) \(\chi_{4033}(1950,\cdot)\) \(\chi_{4033}(1971,\cdot)\) \(\chi_{4033}(2304,\cdot)\) \(\chi_{4033}(2637,\cdot)\) \(\chi_{4033}(2859,\cdot)\) \(\chi_{4033}(3023,\cdot)\) \(\chi_{4033}(3414,\cdot)\) \(\chi_{4033}(3784,\cdot)\)
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: | \(\Q(\zeta_{27})\) |
| Fixed field: | Number field defined by a degree 27 polynomial |
Values on generators
\((1963,2295)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{23}{27}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 4033 }(2637, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{19}{27}\right)\) |
sage: chi.jacobi_sum(n)