Properties

Label 5160.4013
Modulus $5160$
Conductor $5160$
Order $84$
Real no
Primitive yes
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(5160, base_ring=CyclotomicField(84)) M = H._module chi = DirichletCharacter(H, M([0,42,42,63,40]))
 
Copy content pari:[g,chi] = znchar(Mod(4013,5160))
 

Basic properties

Modulus: \(5160\)
Conductor: \(5160\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(84\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: yes
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 5160.gz

\(\chi_{5160}(53,\cdot)\) \(\chi_{5160}(197,\cdot)\) \(\chi_{5160}(533,\cdot)\) \(\chi_{5160}(797,\cdot)\) \(\chi_{5160}(917,\cdot)\) \(\chi_{5160}(1013,\cdot)\) \(\chi_{5160}(1133,\cdot)\) \(\chi_{5160}(1373,\cdot)\) \(\chi_{5160}(1493,\cdot)\) \(\chi_{5160}(1733,\cdot)\) \(\chi_{5160}(1973,\cdot)\) \(\chi_{5160}(2117,\cdot)\) \(\chi_{5160}(2597,\cdot)\) \(\chi_{5160}(2933,\cdot)\) \(\chi_{5160}(3077,\cdot)\) \(\chi_{5160}(3197,\cdot)\) \(\chi_{5160}(3293,\cdot)\) \(\chi_{5160}(3437,\cdot)\) \(\chi_{5160}(3557,\cdot)\) \(\chi_{5160}(3797,\cdot)\) \(\chi_{5160}(3893,\cdot)\) \(\chi_{5160}(4013,\cdot)\) \(\chi_{5160}(4037,\cdot)\) \(\chi_{5160}(4997,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((3871,2581,1721,3097,4561)\) → \((1,-1,-1,-i,e\left(\frac{10}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 5160 }(4013, a) \) \(1\)\(1\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{83}{84}\right)\)\(e\left(\frac{29}{84}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{31}{84}\right)\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{5}{14}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 5160 }(4013,a) \;\) at \(\;a = \) e.g. 2