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Properties

Label 840.17
Modulus $840$
Conductor $105$
Order $12$
Real no
Primitive no
Minimal yes
Parity odd

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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([0,0,6,3,2]))
 
Copy content pari:[g,chi] = znchar(Mod(17,840))
 

Basic properties

Modulus: \(840\)
Conductor: \(105\)
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 \(\chi_{105}(17,\cdot)\)
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.dn

\(\chi_{840}(17,\cdot)\) \(\chi_{840}(257,\cdot)\) \(\chi_{840}(353,\cdot)\) \(\chi_{840}(593,\cdot)\)

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: 12.0.402196204142578125.1

Values on generators

\((631,421,281,337,241)\) → \((1,1,-1,i,e\left(\frac{1}{6}\right))\)

First values

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

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 840 }(17,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 840 }(17,·),\chi_{ 840 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 840 }(17,·)) \;\) at \(\; a,b = \) e.g. 1,2