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Properties

Label 113.41
Modulus $113$
Conductor $113$
Order $56$
Real no
Primitive yes
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(113, base_ring=CyclotomicField(56)) M = H._module chi = DirichletCharacter(H, M([47]))
 
Copy content pari:[g,chi] = znchar(Mod(41,113))
 

Basic properties

Modulus: \(113\)
Conductor: \(113\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(56\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: yes
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 113.i

\(\chi_{113}(9,\cdot)\) \(\chi_{113}(11,\cdot)\) \(\chi_{113}(13,\cdot)\) \(\chi_{113}(22,\cdot)\) \(\chi_{113}(25,\cdot)\) \(\chi_{113}(26,\cdot)\) \(\chi_{113}(31,\cdot)\) \(\chi_{113}(36,\cdot)\) \(\chi_{113}(41,\cdot)\) \(\chi_{113}(50,\cdot)\) \(\chi_{113}(51,\cdot)\) \(\chi_{113}(52,\cdot)\) \(\chi_{113}(61,\cdot)\) \(\chi_{113}(62,\cdot)\) \(\chi_{113}(63,\cdot)\) \(\chi_{113}(72,\cdot)\) \(\chi_{113}(77,\cdot)\) \(\chi_{113}(82,\cdot)\) \(\chi_{113}(87,\cdot)\) \(\chi_{113}(88,\cdot)\) \(\chi_{113}(91,\cdot)\) \(\chi_{113}(100,\cdot)\) \(\chi_{113}(102,\cdot)\) \(\chi_{113}(104,\cdot)\)

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: $\Q(\zeta_{56})$
Fixed field: Number field defined by a degree 56 polynomial

Values on generators

\(3\) → \(e\left(\frac{47}{56}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 113 }(41, a) \) \(1\)\(1\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{47}{56}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{37}{56}\right)\)\(e\left(\frac{51}{56}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{19}{28}\right)\)\(e\left(\frac{41}{56}\right)\)\(e\left(\frac{3}{28}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 113 }(41,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 113 }(41,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 113 }(41,·),\chi_{ 113 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 113 }(41,·)) \;\) at \(\; a,b = \) e.g. 1,2