from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4033, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([42,14]))
pari: [g,chi] = znchar(Mod(2269,4033))
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.dw
\(\chi_{4033}(118,\cdot)\) \(\chi_{4033}(1422,\cdot)\) \(\chi_{4033}(1533,\cdot)\) \(\chi_{4033}(1551,\cdot)\) \(\chi_{4033}(1825,\cdot)\) \(\chi_{4033}(1931,\cdot)\) \(\chi_{4033}(1933,\cdot)\) \(\chi_{4033}(2010,\cdot)\) \(\chi_{4033}(2106,\cdot)\) \(\chi_{4033}(2253,\cdot)\) \(\chi_{4033}(2269,\cdot)\) \(\chi_{4033}(2495,\cdot)\) \(\chi_{4033}(2856,\cdot)\) \(\chi_{4033}(2883,\cdot)\) \(\chi_{4033}(2969,\cdot)\) \(\chi_{4033}(3067,\cdot)\) \(\chi_{4033}(3400,\cdot)\) \(\chi_{4033}(3709,\cdot)\)
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
\((1963,2295)\) → \((e\left(\frac{7}{9}\right),e\left(\frac{7}{27}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 4033 }(2269, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{23}{27}\right)\) |
sage: chi.jacobi_sum(n)