Properties

Label 9600.43
Modulus $9600$
Conductor $640$
Order $32$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9600, base_ring=CyclotomicField(32))
 
M = H._module
 
chi = DirichletCharacter(H, M([16,29,0,24]))
 
pari: [g,chi] = znchar(Mod(43,9600))
 

Basic properties

Modulus: \(9600\)
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 \(\chi_{640}(43,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9600.ew

\(\chi_{9600}(43,\cdot)\) \(\chi_{9600}(307,\cdot)\) \(\chi_{9600}(1243,\cdot)\) \(\chi_{9600}(1507,\cdot)\) \(\chi_{9600}(2443,\cdot)\) \(\chi_{9600}(2707,\cdot)\) \(\chi_{9600}(3643,\cdot)\) \(\chi_{9600}(3907,\cdot)\) \(\chi_{9600}(4843,\cdot)\) \(\chi_{9600}(5107,\cdot)\) \(\chi_{9600}(6043,\cdot)\) \(\chi_{9600}(6307,\cdot)\) \(\chi_{9600}(7243,\cdot)\) \(\chi_{9600}(7507,\cdot)\) \(\chi_{9600}(8443,\cdot)\) \(\chi_{9600}(8707,\cdot)\)

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: 32.32.187072209578355573530071658587684226515959365500928000000000000000000000000.1

Values on generators

\((4351,901,6401,5377)\) → \((-1,e\left(\frac{29}{32}\right),1,-i)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 9600 }(43, a) \) \(1\)\(1\)\(e\left(\frac{5}{16}\right)\)\(e\left(\frac{17}{32}\right)\)\(e\left(\frac{27}{32}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{27}{32}\right)\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{31}{32}\right)\)\(-i\)\(e\left(\frac{13}{32}\right)\)\(e\left(\frac{3}{16}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9600 }(43,a) \;\) at \(\;a = \) e.g. 2