Properties

Label 431.220
Modulus 431431
Conductor 431431
Order 4343
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(431, base_ring=CyclotomicField(86)) M = H._module chi = DirichletCharacter(H, M([68]))
 
Copy content pari:[g,chi] = znchar(Mod(220,431))
 

Basic properties

Modulus: 431431
Conductor: 431431
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: 4343
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 431.e

χ431(2,)\chi_{431}(2,\cdot) χ431(3,)\chi_{431}(3,\cdot) χ431(4,)\chi_{431}(4,\cdot) χ431(6,)\chi_{431}(6,\cdot) χ431(8,)\chi_{431}(8,\cdot) χ431(9,)\chi_{431}(9,\cdot) χ431(12,)\chi_{431}(12,\cdot) χ431(16,)\chi_{431}(16,\cdot) χ431(18,)\chi_{431}(18,\cdot) χ431(24,)\chi_{431}(24,\cdot) χ431(27,)\chi_{431}(27,\cdot) χ431(32,)\chi_{431}(32,\cdot) χ431(36,)\chi_{431}(36,\cdot) χ431(48,)\chi_{431}(48,\cdot) χ431(54,)\chi_{431}(54,\cdot) χ431(55,)\chi_{431}(55,\cdot) χ431(64,)\chi_{431}(64,\cdot) χ431(72,)\chi_{431}(72,\cdot) χ431(81,)\chi_{431}(81,\cdot) χ431(96,)\chi_{431}(96,\cdot) χ431(108,)\chi_{431}(108,\cdot) χ431(110,)\chi_{431}(110,\cdot) χ431(128,)\chi_{431}(128,\cdot) χ431(144,)\chi_{431}(144,\cdot) χ431(145,)\chi_{431}(145,\cdot) χ431(149,)\chi_{431}(149,\cdot) χ431(162,)\chi_{431}(162,\cdot) χ431(165,)\chi_{431}(165,\cdot) χ431(192,)\chi_{431}(192,\cdot) χ431(216,)\chi_{431}(216,\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(ζ43)\Q(\zeta_{43})
Fixed field: Number field defined by a degree 43 polynomial

Values on generators

77e(3443)e\left(\frac{34}{43}\right)

First values

aa 1-111223344556677889910101111
χ431(220,a) \chi_{ 431 }(220, a) 1111e(4143)e\left(\frac{41}{43}\right)e(1743)e\left(\frac{17}{43}\right)e(3943)e\left(\frac{39}{43}\right)e(1843)e\left(\frac{18}{43}\right)e(1543)e\left(\frac{15}{43}\right)e(3443)e\left(\frac{34}{43}\right)e(3743)e\left(\frac{37}{43}\right)e(3443)e\left(\frac{34}{43}\right)e(1643)e\left(\frac{16}{43}\right)e(2243)e\left(\frac{22}{43}\right)
Copy content sage:chi.jacobi_sum(n)
 
χ431(220,a)   \chi_{ 431 }(220,a) \; at   a=\;a = e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
τa(χ431(220,))   \tau_{ a }( \chi_{ 431 }(220,·) )\; at   a=\;a = e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
J(χ431(220,),χ431(n,))   J(\chi_{ 431 }(220,·),\chi_{ 431 }(n,·)) \; for   n= \; n = e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
K(a,b,χ431(220,))  K(a,b,\chi_{ 431 }(220,·)) \; at   a,b=\; a,b = e.g. 1,2