Properties

Label 25.13
Modulus $25$
Conductor $25$
Order $20$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(25, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([19]))
 
pari: [g,chi] = znchar(Mod(13,25))
 

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

Related number fields

Field of values: \(\Q(\zeta_{20})\)
Fixed field: Number field defined by a degree 20 polynomial

Values on generators

\(2\) → \(e\left(\frac{19}{20}\right)\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 25 }(13, a) \) \(-1\)\(1\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{3}{5}\right)\)\(-i\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{1}{20}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 25 }(13,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 25 }(13,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 25 }(13,·),\chi_{ 25 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 25 }(13,·)) \;\) at \(\; a,b = \) e.g. 1,2