Properties

Label 529.48
Modulus 529529
Conductor 529529
Order 253253
Real no
Primitive yes
Minimal yes
Parity even

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(529, base_ring=CyclotomicField(506)) M = H._module chi = DirichletCharacter(H, M([310]))
 
Copy content pari:[g,chi] = znchar(Mod(48,529))
 

Basic properties

Modulus: 529529
Conductor: 529529
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: 253253
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: yes
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 529.g

χ529(2,)\chi_{529}(2,\cdot) χ529(3,)\chi_{529}(3,\cdot) χ529(4,)\chi_{529}(4,\cdot) χ529(6,)\chi_{529}(6,\cdot) χ529(8,)\chi_{529}(8,\cdot) χ529(9,)\chi_{529}(9,\cdot) χ529(12,)\chi_{529}(12,\cdot) χ529(13,)\chi_{529}(13,\cdot) χ529(16,)\chi_{529}(16,\cdot) χ529(18,)\chi_{529}(18,\cdot) χ529(25,)\chi_{529}(25,\cdot) χ529(26,)\chi_{529}(26,\cdot) χ529(27,)\chi_{529}(27,\cdot) χ529(29,)\chi_{529}(29,\cdot) χ529(31,)\chi_{529}(31,\cdot) χ529(32,)\chi_{529}(32,\cdot) χ529(35,)\chi_{529}(35,\cdot) χ529(36,)\chi_{529}(36,\cdot) χ529(39,)\chi_{529}(39,\cdot) χ529(41,)\chi_{529}(41,\cdot) χ529(48,)\chi_{529}(48,\cdot) χ529(49,)\chi_{529}(49,\cdot) χ529(50,)\chi_{529}(50,\cdot) χ529(52,)\chi_{529}(52,\cdot) χ529(54,)\chi_{529}(54,\cdot) χ529(55,)\chi_{529}(55,\cdot) χ529(58,)\chi_{529}(58,\cdot) χ529(59,)\chi_{529}(59,\cdot) χ529(62,)\chi_{529}(62,\cdot) χ529(64,)\chi_{529}(64,\cdot) ...

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

Related number fields

Field of values: Q(ζ253)\Q(\zeta_{253})
Fixed field: Number field defined by a degree 253 polynomial (not computed)

Values on generators

55e(155253)e\left(\frac{155}{253}\right)

First values

aa 1-111223344556677889910101111
χ529(48,a) \chi_{ 529 }(48, a) 1111e(134253)e\left(\frac{134}{253}\right)e(203253)e\left(\frac{203}{253}\right)e(15253)e\left(\frac{15}{253}\right)e(155253)e\left(\frac{155}{253}\right)e(84253)e\left(\frac{84}{253}\right)e(8253)e\left(\frac{8}{253}\right)e(149253)e\left(\frac{149}{253}\right)e(153253)e\left(\frac{153}{253}\right)e(36253)e\left(\frac{36}{253}\right)e(20253)e\left(\frac{20}{253}\right)
Copy content sage:chi.jacobi_sum(n)
 
χ529(48,a)   \chi_{ 529 }(48,a) \; at   a=\;a = e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
τa(χ529(48,))   \tau_{ a }( \chi_{ 529 }(48,·) )\; at   a=\;a = e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
J(χ529(48,),χ529(n,))   J(\chi_{ 529 }(48,·),\chi_{ 529 }(n,·)) \; for   n= \; n = e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
K(a,b,χ529(48,))  K(a,b,\chi_{ 529 }(48,·)) \; at   a,b=\; a,b = e.g. 1,2