Properties

Label 28.11
Modulus $28$
Conductor $28$
Order $6$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(28, base_ring=CyclotomicField(6))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([3,4]))
 
pari: [g,chi] = znchar(Mod(11,28))
 

Basic properties

Modulus: \(28\)
Conductor: \(28\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(6\)
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 28.g

\(\chi_{28}(11,\cdot)\) \(\chi_{28}(23,\cdot)\)

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

Related number fields

Field of values: \(\Q(\sqrt{-3}) \)
Fixed field: 6.0.153664.1

Values on generators

\((15,17)\) → \((-1,e\left(\frac{2}{3}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\(-1\)\(1\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{6}\right)\)\(1\)\(-1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{2}{3}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 28 }(11,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{28}(11,\cdot)) = \sum_{r\in \Z/28\Z} \chi_{28}(11,r) e\left(\frac{r}{14}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 28 }(11,·),\chi_{ 28 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{28}(11,\cdot),\chi_{28}(1,\cdot)) = \sum_{r\in \Z/28\Z} \chi_{28}(11,r) \chi_{28}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 28 }(11,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{28}(11,·)) = \sum_{r \in \Z/28\Z} \chi_{28}(11,r) e\left(\frac{1 r + 2 r^{-1}}{28}\right) = -2.9306671564+1.6920214716i \)