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Properties

Label 3825.2926
Modulus $3825$
Conductor $17$
Order $8$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3825, base_ring=CyclotomicField(8)) M = H._module chi = DirichletCharacter(H, M([0,0,7]))
 
Copy content pari:[g,chi] = znchar(Mod(2926,3825))
 

Basic properties

Modulus: \(3825\)
Conductor: \(17\)
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 \(\chi_{17}(2,\cdot)\)
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 3825.bf

\(\chi_{3825}(451,\cdot)\) \(\chi_{3825}(1351,\cdot)\) \(\chi_{3825}(2701,\cdot)\) \(\chi_{3825}(2926,\cdot)\)

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: \(\Q(\zeta_{17})^+\)

Values on generators

\((2126,2602,2026)\) → \((1,1,e\left(\frac{7}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(19\)\(22\)
\( \chi_{ 3825 }(2926, a) \) \(1\)\(1\)\(i\)\(-1\)\(e\left(\frac{5}{8}\right)\)\(-i\)\(e\left(\frac{1}{8}\right)\)\(-1\)\(e\left(\frac{7}{8}\right)\)\(1\)\(i\)\(e\left(\frac{3}{8}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 3825 }(2926,a) \;\) at \(\;a = \) e.g. 2