Properties

Label 588.223
Modulus $588$
Conductor $196$
Order $14$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(588, base_ring=CyclotomicField(14))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([7,0,1]))
 
pari: [g,chi] = znchar(Mod(223,588))
 

Basic properties

Modulus: \(588\)
Conductor: \(196\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(14\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{196}(27,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 588.x

\(\chi_{588}(55,\cdot)\) \(\chi_{588}(139,\cdot)\) \(\chi_{588}(223,\cdot)\) \(\chi_{588}(307,\cdot)\) \(\chi_{588}(475,\cdot)\) \(\chi_{588}(559,\cdot)\)

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: \(\Q(\zeta_{7})\)
Fixed field: 14.14.21972068264574400934821888.1

Values on generators

\((295,197,493)\) → \((-1,1,e\left(\frac{1}{14}\right))\)

Values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 588 }(223, a) \) \(1\)\(1\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{11}{14}\right)\)\(1\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{2}{7}\right)\)\(1\)\(e\left(\frac{2}{7}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 588 }(223,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 588 }(223,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 588 }(223,·),\chi_{ 588 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 588 }(223,·)) \;\) at \(\; a,b = \) e.g. 1,2