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Properties

Label 3234.961
Modulus $3234$
Conductor $77$
Order $15$
Real no
Primitive no
Minimal no
Parity even

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3234, base_ring=CyclotomicField(30)) M = H._module chi = DirichletCharacter(H, M([0,10,6]))
 
Copy content pari:[g,chi] = znchar(Mod(961,3234))
 

Basic properties

Modulus: \(3234\)
Conductor: \(77\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(15\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{77}(37,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 3234.bg

\(\chi_{3234}(361,\cdot)\) \(\chi_{3234}(949,\cdot)\) \(\chi_{3234}(961,\cdot)\) \(\chi_{3234}(1549,\cdot)\) \(\chi_{3234}(1831,\cdot)\) \(\chi_{3234}(2137,\cdot)\) \(\chi_{3234}(3007,\cdot)\) \(\chi_{3234}(3019,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: 15.15.886528337182930278529.1

Values on generators

\((1079,199,2059)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{1}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 3234 }(961, a) \) \(1\)\(1\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{3}{5}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 3234 }(961,a) \;\) at \(\;a = \) e.g. 2