from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3024, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,27,22,18]))
pari: [g,chi] = znchar(Mod(1805,3024))
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.gv
\(\chi_{3024}(293,\cdot)\) \(\chi_{3024}(461,\cdot)\) \(\chi_{3024}(797,\cdot)\) \(\chi_{3024}(965,\cdot)\) \(\chi_{3024}(1301,\cdot)\) \(\chi_{3024}(1469,\cdot)\) \(\chi_{3024}(1805,\cdot)\) \(\chi_{3024}(1973,\cdot)\) \(\chi_{3024}(2309,\cdot)\) \(\chi_{3024}(2477,\cdot)\) \(\chi_{3024}(2813,\cdot)\) \(\chi_{3024}(2981,\cdot)\)
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: | 36.36.9008390745615103400556966960681548037381304055338896235706938737549613032765449450815488.1 |
Values on generators
\((1135,757,785,2593)\) → \((1,-i,e\left(\frac{11}{18}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 3024 }(1805, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{5}{12}\right)\) |
sage: chi.jacobi_sum(n)