from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1280, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,19,8]))
pari: [g,chi] = znchar(Mod(137,1280))
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 \(\chi_{640}(317,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1280.bq
\(\chi_{1280}(137,\cdot)\) \(\chi_{1280}(153,\cdot)\) \(\chi_{1280}(297,\cdot)\) \(\chi_{1280}(313,\cdot)\) \(\chi_{1280}(457,\cdot)\) \(\chi_{1280}(473,\cdot)\) \(\chi_{1280}(617,\cdot)\) \(\chi_{1280}(633,\cdot)\) \(\chi_{1280}(777,\cdot)\) \(\chi_{1280}(793,\cdot)\) \(\chi_{1280}(937,\cdot)\) \(\chi_{1280}(953,\cdot)\) \(\chi_{1280}(1097,\cdot)\) \(\chi_{1280}(1113,\cdot)\) \(\chi_{1280}(1257,\cdot)\) \(\chi_{1280}(1273,\cdot)\)
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: | 32.0.187072209578355573530071658587684226515959365500928000000000000000000000000.1 |
Values on generators
\((511,261,257)\) → \((1,e\left(\frac{19}{32}\right),i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 1280 }(137, a) \) | \(-1\) | \(1\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{19}{32}\right)\) |
sage: chi.jacobi_sum(n)