from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1656, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,33,55,42]))
pari: [g,chi] = znchar(Mod(59,1656))
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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1656.cf
\(\chi_{1656}(59,\cdot)\) \(\chi_{1656}(131,\cdot)\) \(\chi_{1656}(347,\cdot)\) \(\chi_{1656}(371,\cdot)\) \(\chi_{1656}(443,\cdot)\) \(\chi_{1656}(491,\cdot)\) \(\chi_{1656}(515,\cdot)\) \(\chi_{1656}(587,\cdot)\) \(\chi_{1656}(731,\cdot)\) \(\chi_{1656}(923,\cdot)\) \(\chi_{1656}(947,\cdot)\) \(\chi_{1656}(995,\cdot)\) \(\chi_{1656}(1067,\cdot)\) \(\chi_{1656}(1139,\cdot)\) \(\chi_{1656}(1163,\cdot)\) \(\chi_{1656}(1235,\cdot)\) \(\chi_{1656}(1283,\cdot)\) \(\chi_{1656}(1451,\cdot)\) \(\chi_{1656}(1499,\cdot)\) \(\chi_{1656}(1595,\cdot)\)
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
\((415,829,1289,649)\) → \((-1,-1,e\left(\frac{5}{6}\right),e\left(\frac{7}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 1656 }(59, a) \) | \(1\) | \(1\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{5}{22}\right)\) |
sage: chi.jacobi_sum(n)