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Properties

Label 3040.7
Modulus $3040$
Conductor $1520$
Order $12$
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(3040, base_ring=CyclotomicField(12)) M = H._module chi = DirichletCharacter(H, M([6,3,3,4]))
 
Copy content pari:[g,chi] = znchar(Mod(7,3040))
 

Basic properties

Modulus: \(3040\)
Conductor: \(1520\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(12\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1520}(1147,\cdot)\)
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 3040.cx

\(\chi_{3040}(7,\cdot)\) \(\chi_{3040}(1303,\cdot)\) \(\chi_{3040}(1607,\cdot)\) \(\chi_{3040}(2743,\cdot)\)

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: \(\Q(\zeta_{12})\)
Fixed field: 12.12.284936905588473856000000000.1

Values on generators

\((191,2661,1217,1921)\) → \((-1,i,i,e\left(\frac{1}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 3040 }(7, a) \) \(1\)\(1\)\(e\left(\frac{1}{3}\right)\)\(i\)\(e\left(\frac{2}{3}\right)\)\(-i\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{5}{12}\right)\)\(1\)\(e\left(\frac{11}{12}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 3040 }(7,a) \;\) at \(\;a = \) e.g. 2