from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1152, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,45,40]))
pari: [g,chi] = znchar(Mod(41,1152))
Basic properties
Modulus: | \(1152\) | |
Conductor: | \(576\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{576}(77,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1152.br
\(\chi_{1152}(41,\cdot)\) \(\chi_{1152}(137,\cdot)\) \(\chi_{1152}(185,\cdot)\) \(\chi_{1152}(281,\cdot)\) \(\chi_{1152}(329,\cdot)\) \(\chi_{1152}(425,\cdot)\) \(\chi_{1152}(473,\cdot)\) \(\chi_{1152}(569,\cdot)\) \(\chi_{1152}(617,\cdot)\) \(\chi_{1152}(713,\cdot)\) \(\chi_{1152}(761,\cdot)\) \(\chi_{1152}(857,\cdot)\) \(\chi_{1152}(905,\cdot)\) \(\chi_{1152}(1001,\cdot)\) \(\chi_{1152}(1049,\cdot)\) \(\chi_{1152}(1145,\cdot)\)
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
\((127,901,641)\) → \((1,e\left(\frac{15}{16}\right),e\left(\frac{5}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
\( \chi_{ 1152 }(41, a) \) | \(-1\) | \(1\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(-i\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{1}{6}\right)\) |
sage: chi.jacobi_sum(n)