Properties

Label 588.221
Modulus $588$
Conductor $147$
Order $42$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(588, base_ring=CyclotomicField(42))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,21,16]))
 
pari: [g,chi] = znchar(Mod(221,588))
 

Basic properties

Modulus: \(588\)
Conductor: \(147\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{147}(74,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 588.z

\(\chi_{588}(53,\cdot)\) \(\chi_{588}(65,\cdot)\) \(\chi_{588}(137,\cdot)\) \(\chi_{588}(149,\cdot)\) \(\chi_{588}(221,\cdot)\) \(\chi_{588}(233,\cdot)\) \(\chi_{588}(305,\cdot)\) \(\chi_{588}(317,\cdot)\) \(\chi_{588}(389,\cdot)\) \(\chi_{588}(401,\cdot)\) \(\chi_{588}(473,\cdot)\) \(\chi_{588}(485,\cdot)\)

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: 42.0.176602720807616761537805583365440112858555316650025456145851095351761290003.1

Values on generators

\((295,197,493)\) → \((1,-1,e\left(\frac{8}{21}\right))\)

Values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 588 }(221, a) \) \(-1\)\(1\)\(e\left(\frac{23}{42}\right)\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{41}{42}\right)\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{4}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 588 }(221,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 588 }(221,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 588 }(221,·),\chi_{ 588 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 588 }(221,·)) \;\) at \(\; a,b = \) e.g. 1,2