from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1073, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([27,7]))
pari: [g,chi] = znchar(Mod(17,1073))
Basic properties
| Modulus: | \(1073\) | |
| Conductor: | \(1073\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | 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 1073.bn
\(\chi_{1073}(17,\cdot)\) \(\chi_{1073}(133,\cdot)\) \(\chi_{1073}(220,\cdot)\) \(\chi_{1073}(278,\cdot)\) \(\chi_{1073}(476,\cdot)\) \(\chi_{1073}(505,\cdot)\) \(\chi_{1073}(568,\cdot)\) \(\chi_{1073}(597,\cdot)\) \(\chi_{1073}(795,\cdot)\) \(\chi_{1073}(853,\cdot)\) \(\chi_{1073}(940,\cdot)\) \(\chi_{1073}(1056,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
| Fixed field: | Number field defined by a degree 36 polynomial |
Values on generators
\((408,668)\) → \((-i,e\left(\frac{7}{36}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 1073 }(17, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(-i\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) |
sage: chi.jacobi_sum(n)