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Properties

Label 123.104
Modulus $123$
Conductor $123$
Order $40$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(123, base_ring=CyclotomicField(40)) M = H._module chi = DirichletCharacter(H, M([20,29]))
 
Copy content pari:[g,chi] = znchar(Mod(104,123))
 

Basic properties

Modulus: \(123\)
Conductor: \(123\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(40\)
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 123.o

\(\chi_{123}(11,\cdot)\) \(\chi_{123}(17,\cdot)\) \(\chi_{123}(26,\cdot)\) \(\chi_{123}(29,\cdot)\) \(\chi_{123}(35,\cdot)\) \(\chi_{123}(47,\cdot)\) \(\chi_{123}(53,\cdot)\) \(\chi_{123}(56,\cdot)\) \(\chi_{123}(65,\cdot)\) \(\chi_{123}(71,\cdot)\) \(\chi_{123}(89,\cdot)\) \(\chi_{123}(95,\cdot)\) \(\chi_{123}(101,\cdot)\) \(\chi_{123}(104,\cdot)\) \(\chi_{123}(110,\cdot)\) \(\chi_{123}(116,\cdot)\)

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: \(\Q(\zeta_{40})\)
Fixed field: \(\Q(\zeta_{123})^+\)

Values on generators

\((83,88)\) → \((-1,e\left(\frac{29}{40}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 123 }(104, a) \) \(1\)\(1\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{11}{40}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{27}{40}\right)\)\(e\left(\frac{19}{40}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{2}{5}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 123 }(104,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 123 }(104,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 123 }(104,·),\chi_{ 123 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 123 }(104,·)) \;\) at \(\; a,b = \) e.g. 1,2