from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5733, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,26,14]))
pari: [g,chi] = znchar(Mod(2746,5733))
Basic properties
| Modulus: | \(5733\) | |
| Conductor: | \(637\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(21\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{637}(198,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5733.gw
\(\chi_{5733}(289,\cdot)\) \(\chi_{5733}(1621,\cdot)\) \(\chi_{5733}(1927,\cdot)\) \(\chi_{5733}(2440,\cdot)\) \(\chi_{5733}(2746,\cdot)\) \(\chi_{5733}(3259,\cdot)\) \(\chi_{5733}(3565,\cdot)\) \(\chi_{5733}(4078,\cdot)\) \(\chi_{5733}(4384,\cdot)\) \(\chi_{5733}(4897,\cdot)\) \(\chi_{5733}(5203,\cdot)\) \(\chi_{5733}(5716,\cdot)\)
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
\((2549,1522,5293)\) → \((1,e\left(\frac{13}{21}\right),e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(16\) | \(17\) | \(19\) | \(20\) |
| \( \chi_{ 5733 }(2746, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{17}{21}\right)\) |
sage: chi.jacobi_sum(n)