Properties

Label 700.129
Modulus 700700
Conductor 175175
Order 3030
Real no
Primitive no
Minimal yes
Parity odd

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(700, base_ring=CyclotomicField(30)) M = H._module chi = DirichletCharacter(H, M([0,3,5]))
 
Copy content pari:[g,chi] = znchar(Mod(129,700))
 

Basic properties

Modulus: 700700
Conductor: 175175
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: 3030
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from χ175(129,)\chi_{175}(129,\cdot)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 700.br

χ700(89,)\chi_{700}(89,\cdot) χ700(129,)\chi_{700}(129,\cdot) χ700(229,)\chi_{700}(229,\cdot) χ700(269,)\chi_{700}(269,\cdot) χ700(369,)\chi_{700}(369,\cdot) χ700(409,)\chi_{700}(409,\cdot) χ700(509,)\chi_{700}(509,\cdot) χ700(689,)\chi_{700}(689,\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(ζ15)\Q(\zeta_{15})
Fixed field: 30.0.595554103661284317897450790724178659729659557342529296875.1

Values on generators

(351,477,101)(351,477,101)(1,e(110),e(16))(1,e\left(\frac{1}{10}\right),e\left(\frac{1}{6}\right))

First values

aa 1-111339911111313171719192323272729293131
χ700(129,a) \chi_{ 700 }(129, a) 1-111e(1315)e\left(\frac{13}{15}\right)e(1115)e\left(\frac{11}{15}\right)e(415)e\left(\frac{4}{15}\right)e(25)e\left(\frac{2}{5}\right)e(715)e\left(\frac{7}{15}\right)e(1930)e\left(\frac{19}{30}\right)e(1330)e\left(\frac{13}{30}\right)e(35)e\left(\frac{3}{5}\right)e(15)e\left(\frac{1}{5}\right)e(2930)e\left(\frac{29}{30}\right)
Copy content sage:chi.jacobi_sum(n)
 
χ700(129,a)   \chi_{ 700 }(129,a) \; at   a=\;a = e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
τa(χ700(129,))   \tau_{ a }( \chi_{ 700 }(129,·) )\; at   a=\;a = e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
J(χ700(129,),χ700(n,))   J(\chi_{ 700 }(129,·),\chi_{ 700 }(n,·)) \; for   n= \; n = e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
K(a,b,χ700(129,))  K(a,b,\chi_{ 700 }(129,·)) \; at   a,b=\; a,b = e.g. 1,2