from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1152, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([16,27,0]))
pari: [g,chi] = znchar(Mod(739,1152))
Basic properties
| Modulus: | \(1152\) | |
| Conductor: | \(128\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(32\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{128}(99,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1152.bk
\(\chi_{1152}(19,\cdot)\) \(\chi_{1152}(91,\cdot)\) \(\chi_{1152}(163,\cdot)\) \(\chi_{1152}(235,\cdot)\) \(\chi_{1152}(307,\cdot)\) \(\chi_{1152}(379,\cdot)\) \(\chi_{1152}(451,\cdot)\) \(\chi_{1152}(523,\cdot)\) \(\chi_{1152}(595,\cdot)\) \(\chi_{1152}(667,\cdot)\) \(\chi_{1152}(739,\cdot)\) \(\chi_{1152}(811,\cdot)\) \(\chi_{1152}(883,\cdot)\) \(\chi_{1152}(955,\cdot)\) \(\chi_{1152}(1027,\cdot)\) \(\chi_{1152}(1099,\cdot)\)
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: | 32.0.3138550867693340381917894711603833208051177722232017256448.1 |
Values on generators
\((127,901,641)\) → \((-1,e\left(\frac{27}{32}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 1152 }(739, a) \) | \(-1\) | \(1\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(i\) |
sage: chi.jacobi_sum(n)