from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5077, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([7]))
pari: [g,chi] = znchar(Mod(21,5077))
Basic properties
| Modulus: | \(5077\) | |
| Conductor: | \(5077\) | 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 5077.l
\(\chi_{5077}(21,\cdot)\) \(\chi_{5077}(328,\cdot)\) \(\chi_{5077}(1227,\cdot)\) \(\chi_{5077}(1309,\cdot)\) \(\chi_{5077}(2193,\cdot)\) \(\chi_{5077}(2340,\cdot)\) \(\chi_{5077}(2483,\cdot)\) \(\chi_{5077}(2545,\cdot)\) \(\chi_{5077}(2973,\cdot)\) \(\chi_{5077}(3266,\cdot)\) \(\chi_{5077}(3515,\cdot)\) \(\chi_{5077}(3522,\cdot)\) \(\chi_{5077}(3704,\cdot)\) \(\chi_{5077}(3747,\cdot)\) \(\chi_{5077}(4110,\cdot)\) \(\chi_{5077}(4156,\cdot)\) \(\chi_{5077}(4636,\cdot)\) \(\chi_{5077}(4695,\cdot)\)
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
\(2\) → \(e\left(\frac{7}{54}\right)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 5077 }(21, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{54}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(-1\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{29}{54}\right)\) |
sage: chi.jacobi_sum(n)