from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3024, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,9,8,12]))
pari: [g,chi] = znchar(Mod(1717,3024))
Basic properties
| Modulus: | \(3024\) | |
| Conductor: | \(3024\) | 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 3024.gq
\(\chi_{3024}(205,\cdot)\) \(\chi_{3024}(445,\cdot)\) \(\chi_{3024}(709,\cdot)\) \(\chi_{3024}(949,\cdot)\) \(\chi_{3024}(1213,\cdot)\) \(\chi_{3024}(1453,\cdot)\) \(\chi_{3024}(1717,\cdot)\) \(\chi_{3024}(1957,\cdot)\) \(\chi_{3024}(2221,\cdot)\) \(\chi_{3024}(2461,\cdot)\) \(\chi_{3024}(2725,\cdot)\) \(\chi_{3024}(2965,\cdot)\)
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
\((1135,757,785,2593)\) → \((1,i,e\left(\frac{2}{9}\right),e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 3024 }(1717, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(i\) |
sage: chi.jacobi_sum(n)