from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4033, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([9,49]))
pari: [g,chi] = znchar(Mod(1174,4033))
Basic properties
Modulus: | \(4033\) | |
Conductor: | \(4033\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(54\) | 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.hf
\(\chi_{4033}(249,\cdot)\) \(\chi_{4033}(619,\cdot)\) \(\chi_{4033}(1010,\cdot)\) \(\chi_{4033}(1174,\cdot)\) \(\chi_{4033}(1396,\cdot)\) \(\chi_{4033}(1729,\cdot)\) \(\chi_{4033}(2062,\cdot)\) \(\chi_{4033}(2083,\cdot)\) \(\chi_{4033}(2527,\cdot)\) \(\chi_{4033}(2543,\cdot)\) \(\chi_{4033}(2786,\cdot)\) \(\chi_{4033}(2971,\cdot)\) \(\chi_{4033}(3045,\cdot)\) \(\chi_{4033}(3156,\cdot)\) \(\chi_{4033}(3267,\cdot)\) \(\chi_{4033}(3357,\cdot)\) \(\chi_{4033}(3875,\cdot)\) \(\chi_{4033}(4007,\cdot)\)
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 54 polynomial |
Values on generators
\((1963,2295)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{49}{54}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 4033 }(1174, a) \) | \(1\) | \(1\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{43}{54}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{17}{54}\right)\) |
sage: chi.jacobi_sum(n)