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Properties

Label 420.157
Modulus $420$
Conductor $35$
Order $12$
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(420, base_ring=CyclotomicField(12)) M = H._module chi = DirichletCharacter(H, M([0,0,3,2]))
 
Copy content pari:[g,chi] = znchar(Mod(157,420))
 

Basic properties

Modulus: \(420\)
Conductor: \(35\)
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 \(\chi_{35}(17,\cdot)\)
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 420.bo

\(\chi_{420}(73,\cdot)\) \(\chi_{420}(157,\cdot)\) \(\chi_{420}(313,\cdot)\) \(\chi_{420}(397,\cdot)\)

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: \(\Q(\zeta_{35})^+\)

Values on generators

\((211,281,337,241)\) → \((1,1,i,e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 420 }(157, a) \) \(1\)\(1\)\(e\left(\frac{2}{3}\right)\)\(i\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{12}\right)\)\(-1\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{7}{12}\right)\)\(-1\)\(-i\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 420 }(157,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 420 }(157,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 420 }(157,·),\chi_{ 420 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 420 }(157,·)) \;\) at \(\; a,b = \) e.g. 1,2