Properties

Label 176.147
Modulus 176176
Conductor 176176
Order 2020
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(176, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([10,15,4]))
 
pari: [g,chi] = znchar(Mod(147,176))
 

Basic properties

Modulus: 176176
Conductor: 176176
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: 2020
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 176.v

χ176(3,)\chi_{176}(3,\cdot) χ176(27,)\chi_{176}(27,\cdot) χ176(59,)\chi_{176}(59,\cdot) χ176(75,)\chi_{176}(75,\cdot) χ176(91,)\chi_{176}(91,\cdot) χ176(115,)\chi_{176}(115,\cdot) χ176(147,)\chi_{176}(147,\cdot) χ176(163,)\chi_{176}(163,\cdot)

sage: chi.galois_orbit()
 
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: 20.0.1655513490330868290261743826894848.1

Values on generators

(111,133,145)(111,133,145)(1,i,e(15))(-1,-i,e\left(\frac{1}{5}\right))

First values

aa 1-11133557799131315151717191921212323
χ176(147,a) \chi_{ 176 }(147, a) 1-111e(720)e\left(\frac{7}{20}\right)e(1120)e\left(\frac{11}{20}\right)e(25)e\left(\frac{2}{5}\right)e(710)e\left(\frac{7}{10}\right)e(920)e\left(\frac{9}{20}\right)e(910)e\left(\frac{9}{10}\right)e(45)e\left(\frac{4}{5}\right)e(720)e\left(\frac{7}{20}\right)i-i11
sage: chi.jacobi_sum(n)
 
χ176(147,a)   \chi_{ 176 }(147,a) \; at   a=\;a = e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
τa(χ176(147,))   \tau_{ a }( \chi_{ 176 }(147,·) )\; at   a=\;a = e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
J(χ176(147,),χ176(n,))   J(\chi_{ 176 }(147,·),\chi_{ 176 }(n,·)) \; for   n= \; n = e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
K(a,b,χ176(147,))  K(a,b,\chi_{ 176 }(147,·)) \; at   a,b=\; a,b = e.g. 1,2