Properties

Modulus 52
Conductor 13
Order 12
Real no
Primitive no
Minimal yes
Parity odd
Orbit label 52.k

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(52)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,7]))
 
pari: [g,chi] = znchar(Mod(37,52))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 52
Conductor = 13
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 12
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 = 52.k
Orbit index = 11

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 ]
 

\(\chi_{52}(33,\cdot)\) \(\chi_{52}(37,\cdot)\) \(\chi_{52}(41,\cdot)\) \(\chi_{52}(45,\cdot)\)

Values on generators

\((27,41)\) → \((1,e\left(\frac{7}{12}\right))\)

Values

-113579111517192123
\(-1\)\(1\)\(e\left(\frac{1}{3}\right)\)\(i\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{11}{12}\right)\)\(-i\)\(e\left(\frac{5}{6}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{12})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 52 }(37,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{52}(37,\cdot)) = \sum_{r\in \Z/52\Z} \chi_{52}(37,r) e\left(\frac{r}{26}\right) = -6.1011535954+-3.8439465144i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 52 }(37,·),\chi_{ 52 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{52}(37,\cdot),\chi_{52}(1,\cdot)) = \sum_{r\in \Z/52\Z} \chi_{52}(37,r) \chi_{52}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 52 }(37,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{52}(37,·)) = \sum_{r \in \Z/52\Z} \chi_{52}(37,r) e\left(\frac{1 r + 2 r^{-1}}{52}\right) = 0.0 \)