Properties

Label 26.5
Modulus $26$
Conductor $13$
Order $4$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(26, base_ring=CyclotomicField(4))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([3]))
 
pari: [g,chi] = znchar(Mod(5,26))
 

Basic properties

Modulus: \(26\)
Conductor: \(13\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(4\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{13}(5,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 26.d

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

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

Values on generators

\(15\) → \(-i\)

Values

\(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(15\)\(17\)\(19\)\(21\)\(23\)
\(-1\)\(1\)\(1\)\(-i\)\(i\)\(1\)\(i\)\(-i\)\(-1\)\(-i\)\(i\)\(-1\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\sqrt{-1}) \)
Fixed field: 4.0.2197.1

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
\( \displaystyle K(1,2,\chi_{26}(5,·)) = \sum_{r \in \Z/26\Z} \chi_{26}(5,r) e\left(\frac{1 r + 2 r^{-1}}{26}\right) = -1.473233513+-1.473233513i \)