from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3136, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([12,15,16]))
pari: [g,chi] = znchar(Mod(263,3136))
Basic properties
| Modulus: | \(3136\) | |
| Conductor: | \(224\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(24\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{224}(11,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3136.bs
\(\chi_{3136}(263,\cdot)\) \(\chi_{3136}(471,\cdot)\) \(\chi_{3136}(1047,\cdot)\) \(\chi_{3136}(1255,\cdot)\) \(\chi_{3136}(1831,\cdot)\) \(\chi_{3136}(2039,\cdot)\) \(\chi_{3136}(2615,\cdot)\) \(\chi_{3136}(2823,\cdot)\)
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: | 24.0.329123002999201416128761938882499016916992.1 |
Values on generators
\((1471,197,1473)\) → \((-1,e\left(\frac{5}{8}\right),e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 3136 }(263, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) |
sage: chi.jacobi_sum(n)