Properties

Modulus 98
Conductor 49
Order 42
Real no
Primitive no
Minimal yes
Parity odd
Orbit label 98.h

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(98)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([29]))
 
pari: [g,chi] = znchar(Mod(5,98))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 98
Conductor = 49
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 42
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 = 98.h
Orbit index = 8

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 ]
 

\(\chi_{98}(3,\cdot)\) \(\chi_{98}(5,\cdot)\) \(\chi_{98}(17,\cdot)\) \(\chi_{98}(33,\cdot)\) \(\chi_{98}(45,\cdot)\) \(\chi_{98}(47,\cdot)\) \(\chi_{98}(59,\cdot)\) \(\chi_{98}(61,\cdot)\) \(\chi_{98}(73,\cdot)\) \(\chi_{98}(75,\cdot)\) \(\chi_{98}(87,\cdot)\) \(\chi_{98}(89,\cdot)\)

Values on generators

\(3\) → \(e\left(\frac{29}{42}\right)\)

Values

-1135911131517192325
\(-1\)\(1\)\(e\left(\frac{29}{42}\right)\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{11}{42}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{1}{21}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{21})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 98 }(5,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{98}(5,\cdot)) = \sum_{r\in \Z/98\Z} \chi_{98}(5,r) e\left(\frac{r}{49}\right) = 6.9425300968+0.8951401318i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 98 }(5,·),\chi_{ 98 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{98}(5,\cdot),\chi_{98}(1,\cdot)) = \sum_{r\in \Z/98\Z} \chi_{98}(5,r) \chi_{98}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 98 }(5,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{98}(5,·)) = \sum_{r \in \Z/98\Z} \chi_{98}(5,r) e\left(\frac{1 r + 2 r^{-1}}{98}\right) = 0.0 \)