from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1280, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([16,5,8]))
pari: [g,chi] = znchar(Mod(7,1280))
Basic properties
| Modulus: | \(1280\) | |
| Conductor: | \(640\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(32\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{640}(587,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1280.bl
\(\chi_{1280}(7,\cdot)\) \(\chi_{1280}(23,\cdot)\) \(\chi_{1280}(167,\cdot)\) \(\chi_{1280}(183,\cdot)\) \(\chi_{1280}(327,\cdot)\) \(\chi_{1280}(343,\cdot)\) \(\chi_{1280}(487,\cdot)\) \(\chi_{1280}(503,\cdot)\) \(\chi_{1280}(647,\cdot)\) \(\chi_{1280}(663,\cdot)\) \(\chi_{1280}(807,\cdot)\) \(\chi_{1280}(823,\cdot)\) \(\chi_{1280}(967,\cdot)\) \(\chi_{1280}(983,\cdot)\) \(\chi_{1280}(1127,\cdot)\) \(\chi_{1280}(1143,\cdot)\)
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: | 32.32.187072209578355573530071658587684226515959365500928000000000000000000000000.2 |
Values on generators
\((511,261,257)\) → \((-1,e\left(\frac{5}{32}\right),i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 1280 }(7, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{5}{32}\right)\) |
sage: chi.jacobi_sum(n)