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Properties

Label 153.71
Modulus $153$
Conductor $51$
Order $16$
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(153, base_ring=CyclotomicField(16)) M = H._module chi = DirichletCharacter(H, M([8,1]))
 
Copy content pari:[g,chi] = znchar(Mod(71,153))
 

Basic properties

Modulus: \(153\)
Conductor: \(51\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(16\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{51}(20,\cdot)\)
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 153.o

\(\chi_{153}(44,\cdot)\) \(\chi_{153}(62,\cdot)\) \(\chi_{153}(71,\cdot)\) \(\chi_{153}(80,\cdot)\) \(\chi_{153}(107,\cdot)\) \(\chi_{153}(116,\cdot)\) \(\chi_{153}(125,\cdot)\) \(\chi_{153}(143,\cdot)\)

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: \(\Q(\zeta_{16})\)
Fixed field: \(\Q(\zeta_{51})^+\)

Values on generators

\((137,37)\) → \((-1,e\left(\frac{1}{16}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 153 }(71, a) \) \(1\)\(1\)\(e\left(\frac{3}{8}\right)\)\(-i\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{15}{16}\right)\)\(i\)\(e\left(\frac{1}{16}\right)\)\(-1\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 153 }(71,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 153 }(71,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 153 }(71,·),\chi_{ 153 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 153 }(71,·)) \;\) at \(\; a,b = \) e.g. 1,2