Properties

Label 503.2
Modulus $503$
Conductor $503$
Order $251$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(503, base_ring=CyclotomicField(502))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([202]))
 
pari: [g,chi] = znchar(Mod(2,503))
 

Basic properties

Modulus: \(503\)
Conductor: \(503\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(251\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 503.c

\(\chi_{503}(2,\cdot)\) \(\chi_{503}(3,\cdot)\) \(\chi_{503}(4,\cdot)\) \(\chi_{503}(6,\cdot)\) \(\chi_{503}(7,\cdot)\) \(\chi_{503}(8,\cdot)\) \(\chi_{503}(9,\cdot)\) \(\chi_{503}(11,\cdot)\) \(\chi_{503}(12,\cdot)\) \(\chi_{503}(13,\cdot)\) \(\chi_{503}(14,\cdot)\) \(\chi_{503}(16,\cdot)\) \(\chi_{503}(18,\cdot)\) \(\chi_{503}(21,\cdot)\) \(\chi_{503}(22,\cdot)\) \(\chi_{503}(23,\cdot)\) \(\chi_{503}(24,\cdot)\) \(\chi_{503}(25,\cdot)\) \(\chi_{503}(26,\cdot)\) \(\chi_{503}(27,\cdot)\) \(\chi_{503}(28,\cdot)\) \(\chi_{503}(32,\cdot)\) \(\chi_{503}(33,\cdot)\) \(\chi_{503}(36,\cdot)\) \(\chi_{503}(39,\cdot)\) \(\chi_{503}(42,\cdot)\) \(\chi_{503}(43,\cdot)\) \(\chi_{503}(44,\cdot)\) \(\chi_{503}(46,\cdot)\) \(\chi_{503}(47,\cdot)\) ...

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Values on generators

\(5\) → \(e\left(\frac{101}{251}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{71}{251}\right)\)\(e\left(\frac{194}{251}\right)\)\(e\left(\frac{142}{251}\right)\)\(e\left(\frac{101}{251}\right)\)\(e\left(\frac{14}{251}\right)\)\(e\left(\frac{152}{251}\right)\)\(e\left(\frac{213}{251}\right)\)\(e\left(\frac{137}{251}\right)\)\(e\left(\frac{172}{251}\right)\)\(e\left(\frac{226}{251}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{251})$
Fixed field: Number field defined by a degree 251 polynomial (not computed)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 503 }(2,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{503}(2,\cdot)) = \sum_{r\in \Z/503\Z} \chi_{503}(2,r) e\left(\frac{2r}{503}\right) = -2.8483021926+22.2460597549i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 503 }(2,·),\chi_{ 503 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{503}(2,\cdot),\chi_{503}(1,\cdot)) = \sum_{r\in \Z/503\Z} \chi_{503}(2,r) \chi_{503}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 503 }(2,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{503}(2,·)) = \sum_{r \in \Z/503\Z} \chi_{503}(2,r) e\left(\frac{1 r + 2 r^{-1}}{503}\right) = 4.1103917765+5.060783155i \)