Properties

Label 571.19
Modulus $571$
Conductor $571$
Order $570$
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(570))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([457]))
 
pari: [g,chi] = znchar(Mod(19,571))
 

Basic properties

Modulus: \(571\)
Conductor: \(571\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(570\)
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.p

\(\chi_{571}(3,\cdot)\) \(\chi_{571}(10,\cdot)\) \(\chi_{571}(12,\cdot)\) \(\chi_{571}(17,\cdot)\) \(\chi_{571}(18,\cdot)\) \(\chi_{571}(19,\cdot)\) \(\chi_{571}(28,\cdot)\) \(\chi_{571}(33,\cdot)\) \(\chi_{571}(35,\cdot)\) \(\chi_{571}(39,\cdot)\) \(\chi_{571}(40,\cdot)\) \(\chi_{571}(46,\cdot)\) \(\chi_{571}(53,\cdot)\) \(\chi_{571}(60,\cdot)\) \(\chi_{571}(63,\cdot)\) \(\chi_{571}(67,\cdot)\) \(\chi_{571}(72,\cdot)\) \(\chi_{571}(73,\cdot)\) \(\chi_{571}(74,\cdot)\) \(\chi_{571}(76,\cdot)\) \(\chi_{571}(77,\cdot)\) \(\chi_{571}(79,\cdot)\) \(\chi_{571}(88,\cdot)\) \(\chi_{571}(89,\cdot)\) \(\chi_{571}(91,\cdot)\) \(\chi_{571}(93,\cdot)\) \(\chi_{571}(101,\cdot)\) \(\chi_{571}(102,\cdot)\) \(\chi_{571}(104,\cdot)\) \(\chi_{571}(108,\cdot)\) ...

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: $\Q(\zeta_{285})$
Fixed field: Number field defined by a degree 570 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{457}{570}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(-1\)\(1\)\(e\left(\frac{109}{114}\right)\)\(e\left(\frac{457}{570}\right)\)\(e\left(\frac{52}{57}\right)\)\(e\left(\frac{97}{285}\right)\)\(e\left(\frac{72}{95}\right)\)\(e\left(\frac{39}{190}\right)\)\(e\left(\frac{33}{38}\right)\)\(e\left(\frac{172}{285}\right)\)\(e\left(\frac{169}{570}\right)\)\(e\left(\frac{248}{285}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 571 }(19,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 571 }(19,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 571 }(19,·),\chi_{ 571 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 571 }(19,·)) \;\) at \(\; a,b = \) e.g. 1,2