Properties

Modulus 26
Conductor 13
Order 4
Real no
Primitive no
Minimal yes
Parity odd
Orbit label 26.d

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(26)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([3]))
 
pari: [g,chi] = znchar(Mod(5,26))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 26
Conductor = 13
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 4
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = no
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = odd
Orbit label = 26.d
Orbit index = 4

Galois orbit

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

\(\chi_{26}(5,\cdot)\) \(\chi_{26}(21,\cdot)\)

Values on generators

\(15\) → \(-i\)

Values

-113579111517192123
\(-1\)\(1\)\(1\)\(-i\)\(i\)\(1\)\(i\)\(-i\)\(-1\)\(-i\)\(i\)\(-1\)
value at  e.g. 2

Related number fields

Field of values \(\Q(i)\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 26 }(5,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{26}(5,\cdot)) = \sum_{r\in \Z/26\Z} \chi_{26}(5,r) e\left(\frac{r}{13}\right) = 3.4508443768+1.0448316069i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 26 }(5,·),\chi_{ 26 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{26}(5,\cdot),\chi_{26}(1,\cdot)) = \sum_{r\in \Z/26\Z} \chi_{26}(5,r) \chi_{26}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 26 }(5,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{26}(5,·)) = \sum_{r \in \Z/26\Z} \chi_{26}(5,r) e\left(\frac{1 r + 2 r^{-1}}{26}\right) = -1.473233513+-1.473233513i \)