from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3072, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,11,16]))
pari: [g,chi] = znchar(Mod(161,3072))
Basic properties
| Modulus: | \(3072\) | |
| Conductor: | \(384\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(32\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{384}(221,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3072.x
\(\chi_{3072}(161,\cdot)\) \(\chi_{3072}(353,\cdot)\) \(\chi_{3072}(545,\cdot)\) \(\chi_{3072}(737,\cdot)\) \(\chi_{3072}(929,\cdot)\) \(\chi_{3072}(1121,\cdot)\) \(\chi_{3072}(1313,\cdot)\) \(\chi_{3072}(1505,\cdot)\) \(\chi_{3072}(1697,\cdot)\) \(\chi_{3072}(1889,\cdot)\) \(\chi_{3072}(2081,\cdot)\) \(\chi_{3072}(2273,\cdot)\) \(\chi_{3072}(2465,\cdot)\) \(\chi_{3072}(2657,\cdot)\) \(\chi_{3072}(2849,\cdot)\) \(\chi_{3072}(3041,\cdot)\)
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: | \(\Q(\zeta_{32})\) |
| Fixed field: | 32.0.135104323545903136978453058557785670637514001130337144105502507008.1 |
Values on generators
\((2047,2053,1025)\) → \((1,e\left(\frac{11}{32}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 3072 }(161, a) \) | \(-1\) | \(1\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(-i\) |
sage: chi.jacobi_sum(n)