from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1280, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,29,8]))
pari: [g,chi] = znchar(Mod(57,1280))
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 \(\chi_{640}(597,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1280.bk
\(\chi_{1280}(57,\cdot)\) \(\chi_{1280}(73,\cdot)\) \(\chi_{1280}(217,\cdot)\) \(\chi_{1280}(233,\cdot)\) \(\chi_{1280}(377,\cdot)\) \(\chi_{1280}(393,\cdot)\) \(\chi_{1280}(537,\cdot)\) \(\chi_{1280}(553,\cdot)\) \(\chi_{1280}(697,\cdot)\) \(\chi_{1280}(713,\cdot)\) \(\chi_{1280}(857,\cdot)\) \(\chi_{1280}(873,\cdot)\) \(\chi_{1280}(1017,\cdot)\) \(\chi_{1280}(1033,\cdot)\) \(\chi_{1280}(1177,\cdot)\) \(\chi_{1280}(1193,\cdot)\)
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: | 32.0.187072209578355573530071658587684226515959365500928000000000000000000000000.2 |
Values on generators
\((511,261,257)\) → \((1,e\left(\frac{29}{32}\right),i)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
\( \chi_{ 1280 }(57, a) \) | \(-1\) | \(1\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{13}{32}\right)\) |
sage: chi.jacobi_sum(n)