from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(33327, base_ring=CyclotomicField(1518))
M = H._module
chi = DirichletCharacter(H, M([759,1265,1170]))
pari: [g,chi] = znchar(Mod(26,33327))
Basic properties
| Modulus: | \(33327\) | |
| Conductor: | \(11109\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(1518\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{11109}(26,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 33327.fu
\(\chi_{33327}(26,\cdot)\) \(\chi_{33327}(215,\cdot)\) \(\chi_{33327}(269,\cdot)\) \(\chi_{33327}(278,\cdot)\) \(\chi_{33327}(395,\cdot)\) \(\chi_{33327}(404,\cdot)\) \(\chi_{33327}(584,\cdot)\) \(\chi_{33327}(593,\cdot)\) \(\chi_{33327}(656,\cdot)\) \(\chi_{33327}(719,\cdot)\) \(\chi_{33327}(836,\cdot)\) \(\chi_{33327}(899,\cdot)\) \(\chi_{33327}(1025,\cdot)\) \(\chi_{33327}(1097,\cdot)\) \(\chi_{33327}(1214,\cdot)\) \(\chi_{33327}(1223,\cdot)\) \(\chi_{33327}(1277,\cdot)\) \(\chi_{33327}(1340,\cdot)\) \(\chi_{33327}(1412,\cdot)\) \(\chi_{33327}(1475,\cdot)\)
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: | $\Q(\zeta_{759})$ |
| Fixed field: | Number field defined by a degree 1518 polynomial (not computed) |
Values on generators
\((25922,9523,10585)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{195}{253}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 33327 }(26, a) \) | \(1\) | \(1\) | \(e\left(\frac{481}{1518}\right)\) | \(e\left(\frac{481}{759}\right)\) | \(e\left(\frac{332}{759}\right)\) | \(e\left(\frac{481}{506}\right)\) | \(e\left(\frac{1145}{1518}\right)\) | \(e\left(\frac{1367}{1518}\right)\) | \(e\left(\frac{477}{506}\right)\) | \(e\left(\frac{203}{759}\right)\) | \(e\left(\frac{520}{759}\right)\) | \(e\left(\frac{1171}{1518}\right)\) |
sage: chi.jacobi_sum(n)