Properties

Label 485.23
Modulus 485485
Conductor 485485
Order 9696
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(485, base_ring=CyclotomicField(96)) M = H._module chi = DirichletCharacter(H, M([72,77]))
 
Copy content pari:[g,chi] = znchar(Mod(23,485))
 

Basic properties

Modulus: 485485
Conductor: 485485
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: 9696
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 485.br

χ485(7,)\chi_{485}(7,\cdot) χ485(13,)\chi_{485}(13,\cdot) χ485(23,)\chi_{485}(23,\cdot) χ485(37,)\chi_{485}(37,\cdot) χ485(57,)\chi_{485}(57,\cdot) χ485(68,)\chi_{485}(68,\cdot) χ485(82,)\chi_{485}(82,\cdot) χ485(83,)\chi_{485}(83,\cdot) χ485(87,)\chi_{485}(87,\cdot) χ485(107,)\chi_{485}(107,\cdot) χ485(112,)\chi_{485}(112,\cdot) χ485(118,)\chi_{485}(118,\cdot) χ485(137,)\chi_{485}(137,\cdot) χ485(138,)\chi_{485}(138,\cdot) χ485(153,)\chi_{485}(153,\cdot) χ485(157,)\chi_{485}(157,\cdot) χ485(173,)\chi_{485}(173,\cdot) χ485(187,)\chi_{485}(187,\cdot) χ485(208,)\chi_{485}(208,\cdot) χ485(223,)\chi_{485}(223,\cdot) χ485(232,)\chi_{485}(232,\cdot) χ485(252,)\chi_{485}(252,\cdot) χ485(268,)\chi_{485}(268,\cdot) χ485(278,)\chi_{485}(278,\cdot) χ485(308,)\chi_{485}(308,\cdot) χ485(317,)\chi_{485}(317,\cdot) χ485(362,)\chi_{485}(362,\cdot) χ485(383,)\chi_{485}(383,\cdot) χ485(393,)\chi_{485}(393,\cdot) χ485(427,)\chi_{485}(427,\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(ζ96)\Q(\zeta_{96})
Fixed field: Number field defined by a degree 96 polynomial

Values on generators

(292,296)(292,296)(i,e(7796))(-i,e\left(\frac{77}{96}\right))

First values

aa 1-11122334466778899111112121313
χ485(23,a) \chi_{ 485 }(23, a) 1111e(148)e\left(\frac{1}{48}\right)e(1948)e\left(\frac{19}{48}\right)e(124)e\left(\frac{1}{24}\right)e(512)e\left(\frac{5}{12}\right)e(5996)e\left(\frac{59}{96}\right)e(116)e\left(\frac{1}{16}\right)e(1924)e\left(\frac{19}{24}\right)e(4748)e\left(\frac{47}{48}\right)e(716)e\left(\frac{7}{16}\right)e(2996)e\left(\frac{29}{96}\right)
Copy content sage:chi.jacobi_sum(n)
 
χ485(23,a)   \chi_{ 485 }(23,a) \; at   a=\;a = e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
τa(χ485(23,))   \tau_{ a }( \chi_{ 485 }(23,·) )\; at   a=\;a = e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
J(χ485(23,),χ485(n,))   J(\chi_{ 485 }(23,·),\chi_{ 485 }(n,·)) \; for   n= \; n = e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
K(a,b,χ485(23,))  K(a,b,\chi_{ 485 }(23,·)) \; at   a,b=\; a,b = e.g. 1,2