Properties

Label 4032.2105
Modulus $4032$
Conductor $672$
Order $24$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4032, base_ring=CyclotomicField(24))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,15,12,20]))
 
pari: [g,chi] = znchar(Mod(2105,4032))
 

Basic properties

Modulus: \(4032\)
Conductor: \(672\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(24\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{672}(341,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4032.ge

\(\chi_{4032}(89,\cdot)\) \(\chi_{4032}(521,\cdot)\) \(\chi_{4032}(1097,\cdot)\) \(\chi_{4032}(1529,\cdot)\) \(\chi_{4032}(2105,\cdot)\) \(\chi_{4032}(2537,\cdot)\) \(\chi_{4032}(3113,\cdot)\) \(\chi_{4032}(3545,\cdot)\)

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: \(\Q(\zeta_{24})\)
Fixed field: 24.24.419957608266393538093113781921531638278568932278272.1

Values on generators

\((127,3781,1793,577)\) → \((1,e\left(\frac{5}{8}\right),-1,e\left(\frac{5}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 4032 }(2105, a) \) \(1\)\(1\)\(e\left(\frac{7}{24}\right)\)\(e\left(\frac{23}{24}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{13}{24}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{7}{24}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4032 }(2105,a) \;\) at \(\;a = \) e.g. 2