from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1152, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,5,16]))
pari: [g,chi] = znchar(Mod(53,1152))
Basic properties
Modulus: | \(1152\) | |
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 \(\chi_{384}(53,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1152.bn
\(\chi_{1152}(53,\cdot)\) \(\chi_{1152}(125,\cdot)\) \(\chi_{1152}(197,\cdot)\) \(\chi_{1152}(269,\cdot)\) \(\chi_{1152}(341,\cdot)\) \(\chi_{1152}(413,\cdot)\) \(\chi_{1152}(485,\cdot)\) \(\chi_{1152}(557,\cdot)\) \(\chi_{1152}(629,\cdot)\) \(\chi_{1152}(701,\cdot)\) \(\chi_{1152}(773,\cdot)\) \(\chi_{1152}(845,\cdot)\) \(\chi_{1152}(917,\cdot)\) \(\chi_{1152}(989,\cdot)\) \(\chi_{1152}(1061,\cdot)\) \(\chi_{1152}(1133,\cdot)\)
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
\((127,901,641)\) → \((1,e\left(\frac{5}{32}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
\( \chi_{ 1152 }(53, a) \) | \(-1\) | \(1\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(i\) |
sage: chi.jacobi_sum(n)