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Properties

Label 340.111
Modulus $340$
Conductor $68$
Order $8$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(340, base_ring=CyclotomicField(8)) M = H._module chi = DirichletCharacter(H, M([4,0,1]))
 
Copy content pari:[g,chi] = znchar(Mod(111,340))
 

Basic properties

Modulus: \(340\)
Conductor: \(68\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(8\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{68}(43,\cdot)\)
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 340.v

\(\chi_{340}(111,\cdot)\) \(\chi_{340}(151,\cdot)\) \(\chi_{340}(291,\cdot)\) \(\chi_{340}(331,\cdot)\)

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: \(\Q(\zeta_{8})\)
Fixed field: 8.0.105046700288.1

Values on generators

\((171,137,241)\) → \((-1,1,e\left(\frac{1}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 340 }(111, a) \) \(-1\)\(1\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{7}{8}\right)\)\(i\)\(e\left(\frac{3}{8}\right)\)\(-1\)\(i\)\(-1\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{5}{8}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 340 }(111,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 340 }(111,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 340 }(111,·),\chi_{ 340 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 340 }(111,·)) \;\) at \(\; a,b = \) e.g. 1,2