↑

Properties

Label 840.37
Modulus $840$
Conductor $280$
Order $12$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(840, base_ring=CyclotomicField(12)) M = H._module chi = DirichletCharacter(H, M([0,6,0,3,4]))
 
Copy content pari:[g,chi] = znchar(Mod(37,840))
 

Basic properties

Modulus: \(840\)
Conductor: \(280\)
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 \(\chi_{280}(37,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 840.dm

\(\chi_{840}(37,\cdot)\) \(\chi_{840}(277,\cdot)\) \(\chi_{840}(373,\cdot)\) \(\chi_{840}(613,\cdot)\)

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: 12.0.2951578112000000000.1

Values on generators

\((631,421,281,337,241)\) → \((1,-1,1,i,e\left(\frac{1}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 840 }(37, a) \) \(-1\)\(1\)\(e\left(\frac{5}{6}\right)\)\(i\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{12}\right)\)\(1\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{5}{12}\right)\)\(1\)\(i\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 840 }(37,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 840 }(37,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 840 }(37,·),\chi_{ 840 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 840 }(37,·)) \;\) at \(\; a,b = \) e.g. 1,2