from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1449, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([22,44,12]))
pari: [g,chi] = znchar(Mod(4,1449))
Basic properties
| Modulus: | \(1449\) | |
| Conductor: | \(1449\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(33\) | 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 1449.bx
\(\chi_{1449}(4,\cdot)\) \(\chi_{1449}(16,\cdot)\) \(\chi_{1449}(142,\cdot)\) \(\chi_{1449}(193,\cdot)\) \(\chi_{1449}(256,\cdot)\) \(\chi_{1449}(331,\cdot)\) \(\chi_{1449}(394,\cdot)\) \(\chi_{1449}(445,\cdot)\) \(\chi_{1449}(508,\cdot)\) \(\chi_{1449}(583,\cdot)\) \(\chi_{1449}(634,\cdot)\) \(\chi_{1449}(646,\cdot)\) \(\chi_{1449}(772,\cdot)\) \(\chi_{1449}(823,\cdot)\) \(\chi_{1449}(886,\cdot)\) \(\chi_{1449}(949,\cdot)\) \(\chi_{1449}(961,\cdot)\) \(\chi_{1449}(1024,\cdot)\) \(\chi_{1449}(1087,\cdot)\) \(\chi_{1449}(1327,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
| Fixed field: | Number field defined by a degree 33 polynomial |
Values on generators
\((1289,829,442)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{2}{3}\right),e\left(\frac{2}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 1449 }(4, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) |
sage: chi.jacobi_sum(n)