from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1339, base_ring=CyclotomicField(34))
M = H._module
chi = DirichletCharacter(H, M([17,20]))
pari: [g,chi] = znchar(Mod(64,1339))
Basic properties
| Modulus: | \(1339\) | |
| Conductor: | \(1339\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(34\) | 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 1339.bg
\(\chi_{1339}(64,\cdot)\) \(\chi_{1339}(116,\cdot)\) \(\chi_{1339}(220,\cdot)\) \(\chi_{1339}(272,\cdot)\) \(\chi_{1339}(285,\cdot)\) \(\chi_{1339}(402,\cdot)\) \(\chi_{1339}(493,\cdot)\) \(\chi_{1339}(545,\cdot)\) \(\chi_{1339}(896,\cdot)\) \(\chi_{1339}(935,\cdot)\) \(\chi_{1339}(961,\cdot)\) \(\chi_{1339}(1039,\cdot)\) \(\chi_{1339}(1091,\cdot)\) \(\chi_{1339}(1130,\cdot)\) \(\chi_{1339}(1156,\cdot)\) \(\chi_{1339}(1312,\cdot)\)
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: | \(\Q(\zeta_{17})\) |
| Fixed field: | Number field defined by a degree 34 polynomial |
Values on generators
\((1237,417)\) → \((-1,e\left(\frac{10}{17}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 1339 }(64, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{34}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{29}{34}\right)\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{13}{34}\right)\) |
sage: chi.jacobi_sum(n)