from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3200, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,27,8]))
pari: [g,chi] = znchar(Mod(157,3200))
Basic properties
| Modulus: | \(3200\) | |
| 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 \(\chi_{640}(157,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3200.ck
\(\chi_{3200}(157,\cdot)\) \(\chi_{3200}(293,\cdot)\) \(\chi_{3200}(557,\cdot)\) \(\chi_{3200}(693,\cdot)\) \(\chi_{3200}(957,\cdot)\) \(\chi_{3200}(1093,\cdot)\) \(\chi_{3200}(1357,\cdot)\) \(\chi_{3200}(1493,\cdot)\) \(\chi_{3200}(1757,\cdot)\) \(\chi_{3200}(1893,\cdot)\) \(\chi_{3200}(2157,\cdot)\) \(\chi_{3200}(2293,\cdot)\) \(\chi_{3200}(2557,\cdot)\) \(\chi_{3200}(2693,\cdot)\) \(\chi_{3200}(2957,\cdot)\) \(\chi_{3200}(3093,\cdot)\)
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
\((1151,901,2177)\) → \((1,e\left(\frac{27}{32}\right),i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 3200 }(157, a) \) | \(-1\) | \(1\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{27}{32}\right)\) |
sage: chi.jacobi_sum(n)