Properties

Label 26.15
Modulus $26$
Conductor $13$
Order $12$
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(12))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([1]))
 
pari: [g,chi] = znchar(Mod(15,26))
 

Basic properties

Modulus: \(26\)
Conductor: \(13\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{13}(2,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 26.f

\(\chi_{26}(7,\cdot)\) \(\chi_{26}(11,\cdot)\) \(\chi_{26}(15,\cdot)\) \(\chi_{26}(19,\cdot)\)

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

\(15\) → \(e\left(\frac{1}{12}\right)\)

Values

\(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(15\)\(17\)\(19\)\(21\)\(23\)
\(-1\)\(1\)\(e\left(\frac{1}{3}\right)\)\(-i\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{5}{12}\right)\)\(i\)\(e\left(\frac{5}{6}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{12})\)
Fixed field: \(\Q(\zeta_{13})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 26 }(15,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{26}(15,\cdot)) = \sum_{r\in \Z/26\Z} \chi_{26}(15,r) e\left(\frac{r}{13}\right) = -3.074972059+1.8826966926i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 26 }(15,·),\chi_{ 26 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{26}(15,\cdot),\chi_{26}(1,\cdot)) = \sum_{r\in \Z/26\Z} \chi_{26}(15,r) \chi_{26}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 26 }(15,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{26}(15,·)) = \sum_{r \in \Z/26\Z} \chi_{26}(15,r) e\left(\frac{1 r + 2 r^{-1}}{26}\right) = -1.2538604788+4.6794710124i \)