Properties

Label 6400.7
Modulus $6400$
Conductor $640$
Order $32$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
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)
 
\( \chi_{ 6400 }(7,a) \;\) at \(\;a = \) e.g. 2