from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1792, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,21,16]))
pari: [g,chi] = znchar(Mod(377,1792))
Basic properties
| Modulus: | \(1792\) | |
| Conductor: | \(896\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(32\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{896}(629,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1792.bm
\(\chi_{1792}(41,\cdot)\) \(\chi_{1792}(153,\cdot)\) \(\chi_{1792}(265,\cdot)\) \(\chi_{1792}(377,\cdot)\) \(\chi_{1792}(489,\cdot)\) \(\chi_{1792}(601,\cdot)\) \(\chi_{1792}(713,\cdot)\) \(\chi_{1792}(825,\cdot)\) \(\chi_{1792}(937,\cdot)\) \(\chi_{1792}(1049,\cdot)\) \(\chi_{1792}(1161,\cdot)\) \(\chi_{1792}(1273,\cdot)\) \(\chi_{1792}(1385,\cdot)\) \(\chi_{1792}(1497,\cdot)\) \(\chi_{1792}(1609,\cdot)\) \(\chi_{1792}(1721,\cdot)\)
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: | 32.0.104303243075213755167445035578915122359095224799654955003407693930037248.1 |
Values on generators
\((1023,1541,1025)\) → \((1,e\left(\frac{21}{32}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 1792 }(377, a) \) | \(-1\) | \(1\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) |
sage: chi.jacobi_sum(n)