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Properties

Label 138.61
Modulus $138$
Conductor $23$
Order $22$
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(138, base_ring=CyclotomicField(22)) M = H._module chi = DirichletCharacter(H, M([0,17]))
 
Copy content pari:[g,chi] = znchar(Mod(61,138))
 

Basic properties

Modulus: \(138\)
Conductor: \(23\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(22\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{23}(15,\cdot)\)
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 138.h

\(\chi_{138}(7,\cdot)\) \(\chi_{138}(19,\cdot)\) \(\chi_{138}(37,\cdot)\) \(\chi_{138}(43,\cdot)\) \(\chi_{138}(61,\cdot)\) \(\chi_{138}(67,\cdot)\) \(\chi_{138}(79,\cdot)\) \(\chi_{138}(97,\cdot)\) \(\chi_{138}(103,\cdot)\) \(\chi_{138}(109,\cdot)\)

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: \(\Q(\zeta_{11})\)
Fixed field: Number field defined by a degree 22 polynomial

Values on generators

\((47,97)\) → \((1,e\left(\frac{17}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 138 }(61, a) \) \(-1\)\(1\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{5}{11}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 138 }(61,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 138 }(61,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 138 }(61,·),\chi_{ 138 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 138 }(61,·)) \;\) at \(\; a,b = \) e.g. 1,2