from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5824, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,9,8,16]))
pari: [g,chi] = znchar(Mod(3,5824))
Basic properties
Modulus: | \(5824\) | |
Conductor: | \(5824\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(48\) | 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 5824.nb
\(\chi_{5824}(3,\cdot)\) \(\chi_{5824}(243,\cdot)\) \(\chi_{5824}(731,\cdot)\) \(\chi_{5824}(971,\cdot)\) \(\chi_{5824}(1459,\cdot)\) \(\chi_{5824}(1699,\cdot)\) \(\chi_{5824}(2187,\cdot)\) \(\chi_{5824}(2427,\cdot)\) \(\chi_{5824}(2915,\cdot)\) \(\chi_{5824}(3155,\cdot)\) \(\chi_{5824}(3643,\cdot)\) \(\chi_{5824}(3883,\cdot)\) \(\chi_{5824}(4371,\cdot)\) \(\chi_{5824}(4611,\cdot)\) \(\chi_{5824}(5099,\cdot)\) \(\chi_{5824}(5339,\cdot)\)
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: | \(\Q(\zeta_{48})\) |
Fixed field: | Number field defined by a degree 48 polynomial |
Values on generators
\((2367,1093,4161,4929)\) → \((-1,e\left(\frac{3}{16}\right),e\left(\frac{1}{6}\right),e\left(\frac{1}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
\( \chi_{ 5824 }(3, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{11}{16}\right)\) |
sage: chi.jacobi_sum(n)