from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1339, base_ring=CyclotomicField(34))
M = H._module
chi = DirichletCharacter(H, M([0,16]))
pari: [g,chi] = znchar(Mod(14,1339))
Basic properties
| Modulus: | \(1339\) | |
| Conductor: | \(103\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(17\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{103}(14,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1339.be
\(\chi_{1339}(14,\cdot)\) \(\chi_{1339}(66,\cdot)\) \(\chi_{1339}(79,\cdot)\) \(\chi_{1339}(196,\cdot)\) \(\chi_{1339}(287,\cdot)\) \(\chi_{1339}(339,\cdot)\) \(\chi_{1339}(690,\cdot)\) \(\chi_{1339}(729,\cdot)\) \(\chi_{1339}(755,\cdot)\) \(\chi_{1339}(833,\cdot)\) \(\chi_{1339}(885,\cdot)\) \(\chi_{1339}(924,\cdot)\) \(\chi_{1339}(950,\cdot)\) \(\chi_{1339}(1106,\cdot)\) \(\chi_{1339}(1197,\cdot)\) \(\chi_{1339}(1249,\cdot)\)
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 17 polynomial |
Values on generators
\((1237,417)\) → \((1,e\left(\frac{8}{17}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 1339 }(14, a) \) | \(1\) | \(1\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{12}{17}\right)\) |
sage: chi.jacobi_sum(n)