from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1449, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([44,33,27]))
pari: [g,chi] = znchar(Mod(34,1449))
Basic properties
| Modulus: | \(1449\) | |
| Conductor: | \(1449\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | 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.cu
\(\chi_{1449}(34,\cdot)\) \(\chi_{1449}(76,\cdot)\) \(\chi_{1449}(97,\cdot)\) \(\chi_{1449}(286,\cdot)\) \(\chi_{1449}(412,\cdot)\) \(\chi_{1449}(454,\cdot)\) \(\chi_{1449}(475,\cdot)\) \(\chi_{1449}(517,\cdot)\) \(\chi_{1449}(580,\cdot)\) \(\chi_{1449}(664,\cdot)\) \(\chi_{1449}(727,\cdot)\) \(\chi_{1449}(769,\cdot)\) \(\chi_{1449}(895,\cdot)\) \(\chi_{1449}(916,\cdot)\) \(\chi_{1449}(958,\cdot)\) \(\chi_{1449}(1042,\cdot)\) \(\chi_{1449}(1147,\cdot)\) \(\chi_{1449}(1210,\cdot)\) \(\chi_{1449}(1399,\cdot)\) \(\chi_{1449}(1420,\cdot)\)
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 66 polynomial |
Values on generators
\((1289,829,442)\) → \((e\left(\frac{2}{3}\right),-1,e\left(\frac{9}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 1449 }(34, a) \) | \(1\) | \(1\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) |
sage: chi.jacobi_sum(n)