↑

Properties

Label 3744.883
Modulus $3744$
Conductor $416$
Order $8$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3744, base_ring=CyclotomicField(8)) M = H._module chi = DirichletCharacter(H, M([4,7,0,4]))
 
Copy content pari:[g,chi] = znchar(Mod(883,3744))
 

Basic properties

Modulus: \(3744\)
Conductor: \(416\)
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 \(\chi_{416}(51,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 3744.eh

\(\chi_{3744}(883,\cdot)\) \(\chi_{3744}(1819,\cdot)\) \(\chi_{3744}(2755,\cdot)\) \(\chi_{3744}(3691,\cdot)\)

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: 8.0.61334280470528.11

Values on generators

\((703,2341,2081,2017)\) → \((-1,e\left(\frac{7}{8}\right),1,-1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 3744 }(883, a) \) \(-1\)\(1\)\(e\left(\frac{3}{8}\right)\)\(-i\)\(e\left(\frac{3}{8}\right)\)\(-1\)\(e\left(\frac{1}{8}\right)\)\(-i\)\(-i\)\(e\left(\frac{5}{8}\right)\)\(1\)\(e\left(\frac{1}{8}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 3744 }(883,a) \;\) at \(\;a = \) e.g. 2