from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5824, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([0,9,4,6]))
pari: [g,chi] = znchar(Mod(73,5824))
Basic properties
| Modulus: | \(5824\) | |
| Conductor: | \(2912\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(24\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2912}(2621,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5824.kp
\(\chi_{5824}(73,\cdot)\) \(\chi_{5824}(1305,\cdot)\) \(\chi_{5824}(1321,\cdot)\) \(\chi_{5824}(2553,\cdot)\) \(\chi_{5824}(2985,\cdot)\) \(\chi_{5824}(4217,\cdot)\) \(\chi_{5824}(4233,\cdot)\) \(\chi_{5824}(5465,\cdot)\)
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: | \(\Q(\zeta_{24})\) |
| Fixed field: | Number field defined by a degree 24 polynomial |
Values on generators
\((2367,1093,4161,4929)\) → \((1,e\left(\frac{3}{8}\right),e\left(\frac{1}{6}\right),i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 5824 }(73, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(-i\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{8}\right)\) |
sage: chi.jacobi_sum(n)