Properties

Label 4400.131
Modulus 44004400
Conductor 44004400
Order 2020
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(4400, base_ring=CyclotomicField(20)) M = H._module chi = DirichletCharacter(H, M([10,15,8,10]))
 
Copy content pari:[g,chi] = znchar(Mod(131,4400))
 

Basic properties

Modulus: 44004400
Conductor: 44004400
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: 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 4400.gg

χ4400(131,)\chi_{4400}(131,\cdot) χ4400(571,)\chi_{4400}(571,\cdot) χ4400(1011,)\chi_{4400}(1011,\cdot) χ4400(1891,)\chi_{4400}(1891,\cdot) χ4400(2331,)\chi_{4400}(2331,\cdot) χ4400(2771,)\chi_{4400}(2771,\cdot) χ4400(3211,)\chi_{4400}(3211,\cdot) χ4400(4091,)\chi_{4400}(4091,\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: Number field defined by a degree 20 polynomial

Values on generators

(2751,3301,177,1201)(2751,3301,177,1201)(1,i,e(25),1)(-1,-i,e\left(\frac{2}{5}\right),-1)

First values

aa 1-1113377991313171719192121232327272929
χ4400(131,a) \chi_{ 4400 }(131, a) 1111e(1120)e\left(\frac{11}{20}\right)1-1e(110)e\left(\frac{1}{10}\right)e(720)e\left(\frac{7}{20}\right)e(710)e\left(\frac{7}{10}\right)e(920)e\left(\frac{9}{20}\right)e(120)e\left(\frac{1}{20}\right)e(25)e\left(\frac{2}{5}\right)e(1320)e\left(\frac{13}{20}\right)e(1120)e\left(\frac{11}{20}\right)
Copy content sage:chi.jacobi_sum(n)
 
χ4400(131,a)   \chi_{ 4400 }(131,a) \; at   a=\;a = e.g. 2