Properties

Label 4608.71
Modulus $4608$
Conductor $768$
Order $64$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4608, base_ring=CyclotomicField(64))
 
M = H._module
 
chi = DirichletCharacter(H, M([32,45,32]))
 
pari: [g,chi] = znchar(Mod(71,4608))
 

Basic properties

Modulus: \(4608\)
Conductor: \(768\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(64\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{768}(491,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4608.bu

\(\chi_{4608}(71,\cdot)\) \(\chi_{4608}(215,\cdot)\) \(\chi_{4608}(359,\cdot)\) \(\chi_{4608}(503,\cdot)\) \(\chi_{4608}(647,\cdot)\) \(\chi_{4608}(791,\cdot)\) \(\chi_{4608}(935,\cdot)\) \(\chi_{4608}(1079,\cdot)\) \(\chi_{4608}(1223,\cdot)\) \(\chi_{4608}(1367,\cdot)\) \(\chi_{4608}(1511,\cdot)\) \(\chi_{4608}(1655,\cdot)\) \(\chi_{4608}(1799,\cdot)\) \(\chi_{4608}(1943,\cdot)\) \(\chi_{4608}(2087,\cdot)\) \(\chi_{4608}(2231,\cdot)\) \(\chi_{4608}(2375,\cdot)\) \(\chi_{4608}(2519,\cdot)\) \(\chi_{4608}(2663,\cdot)\) \(\chi_{4608}(2807,\cdot)\) \(\chi_{4608}(2951,\cdot)\) \(\chi_{4608}(3095,\cdot)\) \(\chi_{4608}(3239,\cdot)\) \(\chi_{4608}(3383,\cdot)\) \(\chi_{4608}(3527,\cdot)\) \(\chi_{4608}(3671,\cdot)\) \(\chi_{4608}(3815,\cdot)\) \(\chi_{4608}(3959,\cdot)\) \(\chi_{4608}(4103,\cdot)\) \(\chi_{4608}(4247,\cdot)\) ...

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: $\Q(\zeta_{64})$
Fixed field: Number field defined by a degree 64 polynomial

Values on generators

\((3583,2053,4097)\) → \((-1,e\left(\frac{45}{64}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 4608 }(71, a) \) \(1\)\(1\)\(e\left(\frac{13}{64}\right)\)\(e\left(\frac{17}{32}\right)\)\(e\left(\frac{49}{64}\right)\)\(e\left(\frac{3}{64}\right)\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{43}{64}\right)\)\(e\left(\frac{27}{32}\right)\)\(e\left(\frac{13}{32}\right)\)\(e\left(\frac{63}{64}\right)\)\(e\left(\frac{1}{8}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4608 }(71,a) \;\) at \(\;a = \) e.g. 2