from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1792, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([16,15,0]))
pari: [g,chi] = znchar(Mod(855,1792))
Basic properties
| Modulus: | \(1792\) | |
| 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 \(\chi_{128}(115,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1792.bl
\(\chi_{1792}(71,\cdot)\) \(\chi_{1792}(183,\cdot)\) \(\chi_{1792}(295,\cdot)\) \(\chi_{1792}(407,\cdot)\) \(\chi_{1792}(519,\cdot)\) \(\chi_{1792}(631,\cdot)\) \(\chi_{1792}(743,\cdot)\) \(\chi_{1792}(855,\cdot)\) \(\chi_{1792}(967,\cdot)\) \(\chi_{1792}(1079,\cdot)\) \(\chi_{1792}(1191,\cdot)\) \(\chi_{1792}(1303,\cdot)\) \(\chi_{1792}(1415,\cdot)\) \(\chi_{1792}(1527,\cdot)\) \(\chi_{1792}(1639,\cdot)\) \(\chi_{1792}(1751,\cdot)\)
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
\((1023,1541,1025)\) → \((-1,e\left(\frac{15}{32}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 1792 }(855, a) \) | \(-1\) | \(1\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) |
sage: chi.jacobi_sum(n)