from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3536, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([12,0,16,21]))
pari: [g,chi] = znchar(Mod(1855,3536))
Basic properties
| Modulus: | \(3536\) | |
| Conductor: | \(884\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(24\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{884}(87,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3536.in
\(\chi_{3536}(399,\cdot)\) \(\chi_{3536}(1199,\cdot)\) \(\chi_{3536}(1647,\cdot)\) \(\chi_{3536}(1855,\cdot)\) \(\chi_{3536}(2031,\cdot)\) \(\chi_{3536}(3103,\cdot)\) \(\chi_{3536}(3279,\cdot)\) \(\chi_{3536}(3487,\cdot)\)
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
\((1327,885,3265,1873)\) → \((-1,1,e\left(\frac{2}{3}\right),e\left(\frac{7}{8}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(19\) | \(21\) | \(23\) | \(25\) |
| \( \chi_{ 3536 }(1855, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(-1\) | \(e\left(\frac{7}{24}\right)\) | \(-i\) |
sage: chi.jacobi_sum(n)