Properties

Label 98.5
Modulus $98$
Conductor $49$
Order $42$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(98, base_ring=CyclotomicField(42))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([29]))
 
pari: [g,chi] = znchar(Mod(5,98))
 

Basic properties

Modulus: \(98\)
Conductor: \(49\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{49}(5,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 98.h

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

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: \(\Q(\zeta_{21})\)
Fixed field: \(\Q(\zeta_{49})\)

Values on generators

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

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\(-1\)\(1\)\(e\left(\frac{29}{42}\right)\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{11}{42}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{1}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 98 }(5,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 98 }(5,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 98 }(5,·),\chi_{ 98 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 98 }(5,·)) \;\) at \(\; a,b = \) e.g. 1,2