Properties

Label 784.47
Modulus $784$
Conductor $196$
Order $42$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(784, base_ring=CyclotomicField(42))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([21,0,5]))
 
pari: [g,chi] = znchar(Mod(47,784))
 

Basic properties

Modulus: \(784\)
Conductor: \(196\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{196}(47,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 784.bp

\(\chi_{784}(47,\cdot)\) \(\chi_{784}(143,\cdot)\) \(\chi_{784}(159,\cdot)\) \(\chi_{784}(255,\cdot)\) \(\chi_{784}(271,\cdot)\) \(\chi_{784}(367,\cdot)\) \(\chi_{784}(383,\cdot)\) \(\chi_{784}(479,\cdot)\) \(\chi_{784}(495,\cdot)\) \(\chi_{784}(591,\cdot)\) \(\chi_{784}(703,\cdot)\) \(\chi_{784}(719,\cdot)\)

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

\((687,197,689)\) → \((-1,1,e\left(\frac{5}{42}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\(1\)\(1\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{19}{42}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{11}{42}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{41}{42}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{19}{21}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{21})\)
Fixed field: \(\Q(\zeta_{196})^+\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 784 }(47,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{784}(47,\cdot)) = \sum_{r\in \Z/784\Z} \chi_{784}(47,r) e\left(\frac{r}{392}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 784 }(47,·),\chi_{ 784 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{784}(47,\cdot),\chi_{784}(1,\cdot)) = \sum_{r\in \Z/784\Z} \chi_{784}(47,r) \chi_{784}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 784 }(47,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{784}(47,·)) = \sum_{r \in \Z/784\Z} \chi_{784}(47,r) e\left(\frac{1 r + 2 r^{-1}}{784}\right) = 0.0 \)