Properties

Label 4224.725
Modulus $4224$
Conductor $4224$
Order $32$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4224, base_ring=CyclotomicField(32))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,29,16,16]))
 
pari: [g,chi] = znchar(Mod(725,4224))
 

Basic properties

Modulus: \(4224\)
Conductor: \(4224\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(32\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4224.cn

\(\chi_{4224}(197,\cdot)\) \(\chi_{4224}(461,\cdot)\) \(\chi_{4224}(725,\cdot)\) \(\chi_{4224}(989,\cdot)\) \(\chi_{4224}(1253,\cdot)\) \(\chi_{4224}(1517,\cdot)\) \(\chi_{4224}(1781,\cdot)\) \(\chi_{4224}(2045,\cdot)\) \(\chi_{4224}(2309,\cdot)\) \(\chi_{4224}(2573,\cdot)\) \(\chi_{4224}(2837,\cdot)\) \(\chi_{4224}(3101,\cdot)\) \(\chi_{4224}(3365,\cdot)\) \(\chi_{4224}(3629,\cdot)\) \(\chi_{4224}(3893,\cdot)\) \(\chi_{4224}(4157,\cdot)\)

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: 32.32.6208007170334900849388551577113100621683540552916899074163477799595412799816204288.1

Values on generators

\((2047,133,1409,3841)\) → \((1,e\left(\frac{29}{32}\right),-1,-1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 4224 }(725, a) \) \(1\)\(1\)\(e\left(\frac{13}{32}\right)\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{3}{32}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{11}{32}\right)\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{15}{32}\right)\)\(i\)\(e\left(\frac{31}{32}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4224 }(725,a) \;\) at \(\;a = \) e.g. 2