from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3024, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,27,28,0]))
pari: [g,chi] = znchar(Mod(211,3024))
Basic properties
Modulus: | \(3024\) | |
Conductor: | \(432\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{432}(211,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3024.gw
\(\chi_{3024}(43,\cdot)\) \(\chi_{3024}(211,\cdot)\) \(\chi_{3024}(547,\cdot)\) \(\chi_{3024}(715,\cdot)\) \(\chi_{3024}(1051,\cdot)\) \(\chi_{3024}(1219,\cdot)\) \(\chi_{3024}(1555,\cdot)\) \(\chi_{3024}(1723,\cdot)\) \(\chi_{3024}(2059,\cdot)\) \(\chi_{3024}(2227,\cdot)\) \(\chi_{3024}(2563,\cdot)\) \(\chi_{3024}(2731,\cdot)\)
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: | 36.0.614667125325361522818798575155151578949632894783197825857500612833312768.1 |
Values on generators
\((1135,757,785,2593)\) → \((-1,-i,e\left(\frac{7}{9}\right),1)\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
\( \chi_{ 3024 }(211, a) \) | \(-1\) | \(1\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{5}{12}\right)\) |
sage: chi.jacobi_sum(n)