↑

Properties

Label 101.45
Modulus $101$
Conductor $101$
Order $50$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(101, base_ring=CyclotomicField(50)) M = H._module chi = DirichletCharacter(H, M([31]))
 
Copy content pari:[g,chi] = znchar(Mod(45,101))
 

Basic properties

Modulus: \(101\)
Conductor: \(101\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(50\)
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 101.h

\(\chi_{101}(4,\cdot)\) \(\chi_{101}(9,\cdot)\) \(\chi_{101}(13,\cdot)\) \(\chi_{101}(20,\cdot)\) \(\chi_{101}(21,\cdot)\) \(\chi_{101}(22,\cdot)\) \(\chi_{101}(23,\cdot)\) \(\chi_{101}(30,\cdot)\) \(\chi_{101}(33,\cdot)\) \(\chi_{101}(43,\cdot)\) \(\chi_{101}(45,\cdot)\) \(\chi_{101}(47,\cdot)\) \(\chi_{101}(49,\cdot)\) \(\chi_{101}(64,\cdot)\) \(\chi_{101}(70,\cdot)\) \(\chi_{101}(76,\cdot)\) \(\chi_{101}(77,\cdot)\) \(\chi_{101}(82,\cdot)\) \(\chi_{101}(85,\cdot)\) \(\chi_{101}(96,\cdot)\)

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: \(\Q(\zeta_{25})\)
Fixed field: Number field defined by a degree 50 polynomial

Values on generators

\(2\) → \(e\left(\frac{31}{50}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 101 }(45, a) \) \(1\)\(1\)\(e\left(\frac{31}{50}\right)\)\(e\left(\frac{39}{50}\right)\)\(e\left(\frac{6}{25}\right)\)\(e\left(\frac{22}{25}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{29}{50}\right)\)\(e\left(\frac{43}{50}\right)\)\(e\left(\frac{14}{25}\right)\)\(-1\)\(e\left(\frac{3}{50}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 101 }(45,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 101 }(45,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 101 }(45,·),\chi_{ 101 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 101 }(45,·)) \;\) at \(\; a,b = \) e.g. 1,2