from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5733, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([14,30,14]))
pari: [g,chi] = znchar(Mod(211,5733))
Basic properties
| Modulus: | \(5733\) | |
| Conductor: | \(5733\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(21\) | 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 5733.hg
\(\chi_{5733}(211,\cdot)\) \(\chi_{5733}(1114,\cdot)\) \(\chi_{5733}(1849,\cdot)\) \(\chi_{5733}(1933,\cdot)\) \(\chi_{5733}(2668,\cdot)\) \(\chi_{5733}(2752,\cdot)\) \(\chi_{5733}(3487,\cdot)\) \(\chi_{5733}(3571,\cdot)\) \(\chi_{5733}(4306,\cdot)\) \(\chi_{5733}(4390,\cdot)\) \(\chi_{5733}(5125,\cdot)\) \(\chi_{5733}(5209,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
| Fixed field: | Number field defined by a degree 21 polynomial |
Values on generators
\((2549,1522,5293)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{5}{7}\right),e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(16\) | \(17\) | \(19\) | \(20\) |
| \( \chi_{ 5733 }(211, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{6}{7}\right)\) |
sage: chi.jacobi_sum(n)