from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1412, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,35]))
pari: [g,chi] = znchar(Mod(1123,1412))
Basic properties
| Modulus: | \(1412\) | |
| Conductor: | \(1412\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1412.r
\(\chi_{1412}(35,\cdot)\) \(\chi_{1412}(135,\cdot)\) \(\chi_{1412}(171,\cdot)\) \(\chi_{1412}(191,\cdot)\) \(\chi_{1412}(207,\cdot)\) \(\chi_{1412}(319,\cdot)\) \(\chi_{1412}(387,\cdot)\) \(\chi_{1412}(499,\cdot)\) \(\chi_{1412}(515,\cdot)\) \(\chi_{1412}(535,\cdot)\) \(\chi_{1412}(571,\cdot)\) \(\chi_{1412}(671,\cdot)\) \(\chi_{1412}(827,\cdot)\) \(\chi_{1412}(971,\cdot)\) \(\chi_{1412}(995,\cdot)\) \(\chi_{1412}(1055,\cdot)\) \(\chi_{1412}(1063,\cdot)\) \(\chi_{1412}(1123,\cdot)\) \(\chi_{1412}(1147,\cdot)\) \(\chi_{1412}(1291,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
| Fixed field: | Number field defined by a degree 44 polynomial |
Values on generators
\((707,709)\) → \((-1,e\left(\frac{35}{44}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 1412 }(1123, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(i\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) |
sage: chi.jacobi_sum(n)