Properties

Label 5160.3157
Modulus $5160$
Conductor $1720$
Order $84$
Real no
Primitive no
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([0,42,0,21,58]))
 
Copy content pari:[g,chi] = znchar(Mod(3157,5160))
 

Basic properties

Modulus: \(5160\)
Conductor: \(1720\)
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: no, induced from \(\chi_{1720}(1437,\cdot)\)
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 5160.hd

\(\chi_{5160}(157,\cdot)\) \(\chi_{5160}(277,\cdot)\) \(\chi_{5160}(373,\cdot)\) \(\chi_{5160}(493,\cdot)\) \(\chi_{5160}(757,\cdot)\) \(\chi_{5160}(1093,\cdot)\) \(\chi_{5160}(1237,\cdot)\) \(\chi_{5160}(1453,\cdot)\) \(\chi_{5160}(2413,\cdot)\) \(\chi_{5160}(2437,\cdot)\) \(\chi_{5160}(2557,\cdot)\) \(\chi_{5160}(2653,\cdot)\) \(\chi_{5160}(2893,\cdot)\) \(\chi_{5160}(3013,\cdot)\) \(\chi_{5160}(3157,\cdot)\) \(\chi_{5160}(3253,\cdot)\) \(\chi_{5160}(3373,\cdot)\) \(\chi_{5160}(3517,\cdot)\) \(\chi_{5160}(3853,\cdot)\) \(\chi_{5160}(4333,\cdot)\) \(\chi_{5160}(4477,\cdot)\) \(\chi_{5160}(4717,\cdot)\) \(\chi_{5160}(4957,\cdot)\) \(\chi_{5160}(5077,\cdot)\)

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

\((3871,2581,1721,3097,4561)\) → \((1,-1,1,i,e\left(\frac{29}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 5160 }(3157, a) \) \(1\)\(1\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{29}{84}\right)\)\(e\left(\frac{41}{84}\right)\)\(e\left(\frac{5}{42}\right)\)\(e\left(\frac{67}{84}\right)\)\(e\left(\frac{13}{42}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{1}{7}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 5160 }(3157,a) \;\) at \(\;a = \) e.g. 2