from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5824, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,27,40,12]))
pari: [g,chi] = znchar(Mod(229,5824))
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.lk
\(\chi_{5824}(229,\cdot)\) \(\chi_{5824}(437,\cdot)\) \(\chi_{5824}(733,\cdot)\) \(\chi_{5824}(941,\cdot)\) \(\chi_{5824}(1685,\cdot)\) \(\chi_{5824}(1893,\cdot)\) \(\chi_{5824}(2189,\cdot)\) \(\chi_{5824}(2397,\cdot)\) \(\chi_{5824}(3141,\cdot)\) \(\chi_{5824}(3349,\cdot)\) \(\chi_{5824}(3645,\cdot)\) \(\chi_{5824}(3853,\cdot)\) \(\chi_{5824}(4597,\cdot)\) \(\chi_{5824}(4805,\cdot)\) \(\chi_{5824}(5101,\cdot)\) \(\chi_{5824}(5309,\cdot)\)
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
\((2367,1093,4161,4929)\) → \((1,e\left(\frac{9}{16}\right),e\left(\frac{5}{6}\right),i)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
\( \chi_{ 5824 }(229, a) \) | \(1\) | \(1\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(-1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{9}{16}\right)\) |
sage: chi.jacobi_sum(n)