from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6400, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,5,16]))
pari: [g,chi] = znchar(Mod(249,6400))
Basic properties
| Modulus: | \(6400\) | |
| 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 \(\chi_{640}(309,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6400.ci
\(\chi_{6400}(249,\cdot)\) \(\chi_{6400}(649,\cdot)\) \(\chi_{6400}(1049,\cdot)\) \(\chi_{6400}(1449,\cdot)\) \(\chi_{6400}(1849,\cdot)\) \(\chi_{6400}(2249,\cdot)\) \(\chi_{6400}(2649,\cdot)\) \(\chi_{6400}(3049,\cdot)\) \(\chi_{6400}(3449,\cdot)\) \(\chi_{6400}(3849,\cdot)\) \(\chi_{6400}(4249,\cdot)\) \(\chi_{6400}(4649,\cdot)\) \(\chi_{6400}(5049,\cdot)\) \(\chi_{6400}(5449,\cdot)\) \(\chi_{6400}(5849,\cdot)\) \(\chi_{6400}(6249,\cdot)\)
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: | 32.32.478904856520590268236983445984471619880855975682375680000000000000000.1 |
Values on generators
\((4351,4101,5377)\) → \((1,e\left(\frac{5}{32}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 6400 }(249, a) \) | \(1\) | \(1\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{29}{32}\right)\) |
sage: chi.jacobi_sum(n)