Properties

Label 571.2
Modulus $571$
Conductor $571$
Order $114$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(571, base_ring=CyclotomicField(114))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([109]))
 
pari: [g,chi] = znchar(Mod(2,571))
 

Basic properties

Modulus: \(571\)
Conductor: \(571\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(114\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 571.m

\(\chi_{571}(2,\cdot)\) \(\chi_{571}(32,\cdot)\) \(\chi_{571}(41,\cdot)\) \(\chi_{571}(47,\cdot)\) \(\chi_{571}(58,\cdot)\) \(\chi_{571}(62,\cdot)\) \(\chi_{571}(75,\cdot)\) \(\chi_{571}(85,\cdot)\) \(\chi_{571}(107,\cdot)\) \(\chi_{571}(118,\cdot)\) \(\chi_{571}(128,\cdot)\) \(\chi_{571}(129,\cdot)\) \(\chi_{571}(135,\cdot)\) \(\chi_{571}(153,\cdot)\) \(\chi_{571}(175,\cdot)\) \(\chi_{571}(188,\cdot)\) \(\chi_{571}(195,\cdot)\) \(\chi_{571}(198,\cdot)\) \(\chi_{571}(209,\cdot)\) \(\chi_{571}(232,\cdot)\) \(\chi_{571}(243,\cdot)\) \(\chi_{571}(262,\cdot)\) \(\chi_{571}(286,\cdot)\) \(\chi_{571}(301,\cdot)\) \(\chi_{571}(313,\cdot)\) \(\chi_{571}(315,\cdot)\) \(\chi_{571}(335,\cdot)\) \(\chi_{571}(340,\cdot)\) \(\chi_{571}(351,\cdot)\) \(\chi_{571}(421,\cdot)\) ...

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: $\Q(\zeta_{57})$
Fixed field: Number field defined by a degree 114 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{109}{114}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(-1\)\(1\)\(e\left(\frac{11}{114}\right)\)\(e\left(\frac{109}{114}\right)\)\(e\left(\frac{11}{57}\right)\)\(e\left(\frac{28}{57}\right)\)\(e\left(\frac{1}{19}\right)\)\(e\left(\frac{33}{38}\right)\)\(e\left(\frac{11}{38}\right)\)\(e\left(\frac{52}{57}\right)\)\(e\left(\frac{67}{114}\right)\)\(e\left(\frac{14}{57}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 571 }(2,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 571 }(2,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 571 }(2,·),\chi_{ 571 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 571 }(2,·)) \;\) at \(\; a,b = \) e.g. 1,2