Properties

Label 11.5
Modulus 1111
Conductor 1111
Order 55
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(11, base_ring=CyclotomicField(10))
 
M = H._module
 
chi = DirichletCharacter(H, M([4]))
 
pari: [g,chi] = znchar(Mod(5,11))
 

Basic properties

Modulus: 1111
Conductor: 1111
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: 55
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 11.c

χ11(3,)\chi_{11}(3,\cdot) χ11(4,)\chi_{11}(4,\cdot) χ11(5,)\chi_{11}(5,\cdot) χ11(9,)\chi_{11}(9,\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(ζ5)\Q(\zeta_{5})
Fixed field: Q(ζ11)+\Q(\zeta_{11})^+

Values on generators

22e(25)e\left(\frac{2}{5}\right)

Values

aa 1-1112233445566778899
χ11(5,a) \chi_{ 11 }(5, a) 1111e(25)e\left(\frac{2}{5}\right)e(15)e\left(\frac{1}{5}\right)e(45)e\left(\frac{4}{5}\right)e(35)e\left(\frac{3}{5}\right)e(35)e\left(\frac{3}{5}\right)e(45)e\left(\frac{4}{5}\right)e(15)e\left(\frac{1}{5}\right)e(25)e\left(\frac{2}{5}\right)
sage: chi.jacobi_sum(n)
 
χ11(5,a)   \chi_{ 11 }(5,a) \; at   a=\;a = e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
τa(χ11(5,))   \tau_{ a }( \chi_{ 11 }(5,·) )\; at   a=\;a = e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
J(χ11(5,),χ11(n,))   J(\chi_{ 11 }(5,·),\chi_{ 11 }(n,·)) \; for   n= \; n = e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
K(a,b,χ11(5,))  K(a,b,\chi_{ 11 }(5,·)) \; at   a,b=\; a,b = e.g. 1,2