Properties

Label 25.8
Modulus $25$
Conductor $25$
Order $20$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(25, base_ring=CyclotomicField(20))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([3]))
 
pari: [g,chi] = znchar(Mod(8,25))
 

Basic properties

Modulus: \(25\)
Conductor: \(25\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 25.f

\(\chi_{25}(2,\cdot)\) \(\chi_{25}(3,\cdot)\) \(\chi_{25}(8,\cdot)\) \(\chi_{25}(12,\cdot)\) \(\chi_{25}(13,\cdot)\) \(\chi_{25}(17,\cdot)\) \(\chi_{25}(22,\cdot)\) \(\chi_{25}(23,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{20})\)
Fixed field: \(\Q(\zeta_{25})\)

Values on generators

\(2\) → \(e\left(\frac{3}{20}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\(-1\)\(1\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{1}{5}\right)\)\(-i\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{17}{20}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 25 }(8,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 25 }(8,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 25 }(8,·),\chi_{ 25 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 25 }(8,·)) \;\) at \(\; a,b = \) e.g. 1,2