from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6400, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([16,5,8]))
pari: [g,chi] = znchar(Mod(7,6400))
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 \(\chi_{640}(587,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6400.cf
\(\chi_{6400}(7,\cdot)\) \(\chi_{6400}(343,\cdot)\) \(\chi_{6400}(807,\cdot)\) \(\chi_{6400}(1143,\cdot)\) \(\chi_{6400}(1607,\cdot)\) \(\chi_{6400}(1943,\cdot)\) \(\chi_{6400}(2407,\cdot)\) \(\chi_{6400}(2743,\cdot)\) \(\chi_{6400}(3207,\cdot)\) \(\chi_{6400}(3543,\cdot)\) \(\chi_{6400}(4007,\cdot)\) \(\chi_{6400}(4343,\cdot)\) \(\chi_{6400}(4807,\cdot)\) \(\chi_{6400}(5143,\cdot)\) \(\chi_{6400}(5607,\cdot)\) \(\chi_{6400}(5943,\cdot)\)
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
\((4351,4101,5377)\) → \((-1,e\left(\frac{5}{32}\right),i)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
\( \chi_{ 6400 }(7, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{5}{32}\right)\) |
sage: chi.jacobi_sum(n)