Properties

Label 440.183
Modulus 440440
Conductor 220220
Order 2020
Real no
Primitive no
Minimal no
Parity odd

Related objects

Downloads

Learn more

Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(440, base_ring=CyclotomicField(20)) M = H._module chi = DirichletCharacter(H, M([10,0,15,14]))
 
Copy content pari:[g,chi] = znchar(Mod(183,440))
 

Basic properties

Modulus: 440440
Conductor: 220220
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: 2020
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from χ220(183,)\chi_{220}(183,\cdot)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 440.bt

χ440(7,)\chi_{440}(7,\cdot) χ440(63,)\chi_{440}(63,\cdot) χ440(127,)\chi_{440}(127,\cdot) χ440(167,)\chi_{440}(167,\cdot) χ440(183,)\chi_{440}(183,\cdot) χ440(303,)\chi_{440}(303,\cdot) χ440(327,)\chi_{440}(327,\cdot) χ440(343,)\chi_{440}(343,\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(ζ20)\Q(\zeta_{20})
Fixed field: 20.0.177917354031751407392000000000000000.1

Values on generators

(111,221,177,321)(111,221,177,321)(1,1,i,e(710))(-1,1,-i,e\left(\frac{7}{10}\right))

First values

aa 1-1113377991313171719192121232327272929
χ440(183,a) \chi_{ 440 }(183, a) 1-111e(720)e\left(\frac{7}{20}\right)e(320)e\left(\frac{3}{20}\right)e(710)e\left(\frac{7}{10}\right)e(1920)e\left(\frac{19}{20}\right)e(120)e\left(\frac{1}{20}\right)e(110)e\left(\frac{1}{10}\right)1-1i-ie(120)e\left(\frac{1}{20}\right)e(25)e\left(\frac{2}{5}\right)
Copy content sage:chi.jacobi_sum(n)
 
χ440(183,a)   \chi_{ 440 }(183,a) \; at   a=\;a = e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
τa(χ440(183,))   \tau_{ a }( \chi_{ 440 }(183,·) )\; at   a=\;a = e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
J(χ440(183,),χ440(n,))   J(\chi_{ 440 }(183,·),\chi_{ 440 }(n,·)) \; for   n= \; n = e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
K(a,b,χ440(183,))  K(a,b,\chi_{ 440 }(183,·)) \; at   a,b=\; a,b = e.g. 1,2