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Properties

Label 34272.4157
Modulus $34272$
Conductor $11424$
Order $8$
Real no
Primitive no
Minimal yes
Parity even

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(34272, base_ring=CyclotomicField(8)) M = H._module chi = DirichletCharacter(H, M([0,3,4,4,1]))
 
Copy content pari:[g,chi] = znchar(Mod(4157,34272))
 

Basic properties

Modulus: \(34272\)
Conductor: \(11424\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(8\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{11424}(4157,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 34272.ln

\(\chi_{34272}(3653,\cdot)\) \(\chi_{34272}(4157,\cdot)\) \(\chi_{34272}(14741,\cdot)\) \(\chi_{34272}(31373,\cdot)\)

Copy content sage:chi.galois_orbit()
 
Copy content pari:order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{8})\)
Fixed field: Number field defined by a degree 8 polynomial

Values on generators

\((2143,29989,3809,14689,14113)\) → \((1,e\left(\frac{3}{8}\right),-1,-1,e\left(\frac{1}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 34272 }(4157, a) \) \(1\)\(1\)\(1\)\(i\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{5}{8}\right)\)\(1\)\(i\)\(e\left(\frac{5}{8}\right)\)\(-1\)\(e\left(\frac{5}{8}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 34272 }(4157,a) \;\) at \(\;a = \) e.g. 2