↑

Properties

Label 103.53
Modulus $103$
Conductor $103$
Order $102$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(103, base_ring=CyclotomicField(102)) M = H._module chi = DirichletCharacter(H, M([97]))
 
Copy content gp:[g,chi] = znchar(Mod(53, 103))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("103.53");
 

Basic properties

Modulus: \(103\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(103\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(102\)
Copy content comment:Order
 
Copy content sage:chi.multiplicative_order()
 
Copy content gp:charorder(g,chi)
 
Copy content magma:Order(chi);
 
Real: no
Copy content comment:Whether the character is real
 
Copy content sage:chi.multiplicative_order() <= 2
 
Copy content gp:charorder(g,chi) <= 2
 
Copy content magma:Order(chi) le 2;
 
Primitive: yes
Copy content comment:If the character is primitive
 
Copy content sage:chi.is_primitive()
 
Copy content gp:#znconreyconductor(g,chi)==1
 
Copy content magma:IsPrimitive(chi);
 
Minimal: yes
Parity: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 103.h

\(\chi_{103}(5,\cdot)\) \(\chi_{103}(6,\cdot)\) \(\chi_{103}(11,\cdot)\) \(\chi_{103}(12,\cdot)\) \(\chi_{103}(20,\cdot)\) \(\chi_{103}(21,\cdot)\) \(\chi_{103}(35,\cdot)\) \(\chi_{103}(40,\cdot)\) \(\chi_{103}(43,\cdot)\) \(\chi_{103}(44,\cdot)\) \(\chi_{103}(45,\cdot)\) \(\chi_{103}(48,\cdot)\) \(\chi_{103}(51,\cdot)\) \(\chi_{103}(53,\cdot)\) \(\chi_{103}(54,\cdot)\) \(\chi_{103}(62,\cdot)\) \(\chi_{103}(65,\cdot)\) \(\chi_{103}(67,\cdot)\) \(\chi_{103}(70,\cdot)\) \(\chi_{103}(71,\cdot)\) \(\chi_{103}(74,\cdot)\) \(\chi_{103}(75,\cdot)\) \(\chi_{103}(77,\cdot)\) \(\chi_{103}(78,\cdot)\) \(\chi_{103}(84,\cdot)\) \(\chi_{103}(85,\cdot)\) \(\chi_{103}(86,\cdot)\) \(\chi_{103}(87,\cdot)\) \(\chi_{103}(88,\cdot)\) \(\chi_{103}(96,\cdot)\) ...

Copy content comment:Galois orbit
 
Copy content sage:chi.galois_orbit()
 
Copy content gp:order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 
Copy content magma:order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Related number fields

Field of values: $\Q(\zeta_{51})$
Fixed field: Number field defined by a degree 102 polynomial (not computed)

Values on generators

\(5\) → \(e\left(\frac{97}{102}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 103 }(53, a) \) \(-1\)\(1\)\(e\left(\frac{43}{51}\right)\)\(e\left(\frac{3}{34}\right)\)\(e\left(\frac{35}{51}\right)\)\(e\left(\frac{97}{102}\right)\)\(e\left(\frac{95}{102}\right)\)\(e\left(\frac{41}{51}\right)\)\(e\left(\frac{9}{17}\right)\)\(e\left(\frac{3}{17}\right)\)\(e\left(\frac{27}{34}\right)\)\(e\left(\frac{1}{102}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x)
 
Copy content gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
 
Copy content magma:chi(x)
 
\( \chi_{ 103 }(53,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content comment:Gauss sum
 
Copy content sage:chi.gauss_sum(a)
 
Copy content gp:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 103 }(53,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content comment:Jacobi sum
 
Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 103 }(53,·),\chi_{ 103 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content comment:Kloosterman sum
 
Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 103 }(53,·)) \;\) at \(\; a,b = \) e.g. 1,2