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Properties

Label 296.113
Modulus $296$
Conductor $37$
Order $36$
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(296, base_ring=CyclotomicField(36)) M = H._module chi = DirichletCharacter(H, M([0,0,1]))
 
Copy content pari:[g,chi] = znchar(Mod(113,296))
 

Basic properties

Modulus: \(296\)
Conductor: \(37\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(36\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{37}(2,\cdot)\)
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 296.bi

\(\chi_{296}(17,\cdot)\) \(\chi_{296}(57,\cdot)\) \(\chi_{296}(89,\cdot)\) \(\chi_{296}(113,\cdot)\) \(\chi_{296}(129,\cdot)\) \(\chi_{296}(153,\cdot)\) \(\chi_{296}(161,\cdot)\) \(\chi_{296}(209,\cdot)\) \(\chi_{296}(217,\cdot)\) \(\chi_{296}(241,\cdot)\) \(\chi_{296}(257,\cdot)\) \(\chi_{296}(281,\cdot)\)

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: \(\Q(\zeta_{36})\)
Fixed field: Number field defined by a degree 36 polynomial

Values on generators

\((223,149,113)\) → \((1,1,e\left(\frac{1}{36}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 296 }(113, a) \) \(-1\)\(1\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{23}{36}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{11}{36}\right)\)\(e\left(\frac{13}{36}\right)\)\(e\left(\frac{7}{36}\right)\)\(e\left(\frac{35}{36}\right)\)\(e\left(\frac{11}{18}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 296 }(113,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 296 }(113,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 296 }(113,·),\chi_{ 296 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 296 }(113,·)) \;\) at \(\; a,b = \) e.g. 1,2