from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1656, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,33,11,27]))
pari: [g,chi] = znchar(Mod(11,1656))
Basic properties
| Modulus: | \(1656\) | |
| Conductor: | \(1656\) | 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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1656.cd
\(\chi_{1656}(11,\cdot)\) \(\chi_{1656}(83,\cdot)\) \(\chi_{1656}(155,\cdot)\) \(\chi_{1656}(203,\cdot)\) \(\chi_{1656}(227,\cdot)\) \(\chi_{1656}(419,\cdot)\) \(\chi_{1656}(563,\cdot)\) \(\chi_{1656}(635,\cdot)\) \(\chi_{1656}(659,\cdot)\) \(\chi_{1656}(707,\cdot)\) \(\chi_{1656}(779,\cdot)\) \(\chi_{1656}(803,\cdot)\) \(\chi_{1656}(1019,\cdot)\) \(\chi_{1656}(1091,\cdot)\) \(\chi_{1656}(1211,\cdot)\) \(\chi_{1656}(1307,\cdot)\) \(\chi_{1656}(1355,\cdot)\) \(\chi_{1656}(1523,\cdot)\) \(\chi_{1656}(1571,\cdot)\) \(\chi_{1656}(1643,\cdot)\)
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
\((415,829,1289,649)\) → \((-1,-1,e\left(\frac{1}{6}\right),e\left(\frac{9}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 1656 }(11, a) \) | \(-1\) | \(1\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{15}{22}\right)\) |
sage: chi.jacobi_sum(n)