from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4033, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([30,7]))
pari: [g,chi] = znchar(Mod(3955,4033))
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.hr
\(\chi_{4033}(145,\cdot)\) \(\chi_{4033}(238,\cdot)\) \(\chi_{4033}(497,\cdot)\) \(\chi_{4033}(682,\cdot)\) \(\chi_{4033}(932,\cdot)\) \(\chi_{4033}(1196,\cdot)\) \(\chi_{4033}(1709,\cdot)\) \(\chi_{4033}(1735,\cdot)\) \(\chi_{4033}(1936,\cdot)\) \(\chi_{4033}(2192,\cdot)\) \(\chi_{4033}(2264,\cdot)\) \(\chi_{4033}(2972,\cdot)\) \(\chi_{4033}(3030,\cdot)\) \(\chi_{4033}(3031,\cdot)\) \(\chi_{4033}(3154,\cdot)\) \(\chi_{4033}(3265,\cdot)\) \(\chi_{4033}(3364,\cdot)\) \(\chi_{4033}(3955,\cdot)\)
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
\((1963,2295)\) → \((e\left(\frac{5}{9}\right),e\left(\frac{7}{54}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 4033 }(3955, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{7}{54}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{23}{54}\right)\) |
sage: chi.jacobi_sum(n)