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Properties

Label 7.4
Modulus 77
Conductor 77
Order 33
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7, base_ring=CyclotomicField(6))
 
M = H._module
 
chi = DirichletCharacter(H, M([4]))
 
pari: [g,chi] = znchar(Mod(4,7))
 

Basic properties

Modulus: 77
Conductor: 77
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: 33
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 7.c

χ7(2,⋅)\chi_{7}(2,\cdot) χ7(4,⋅)\chi_{7}(4,\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(ζ3)\mathbb{Q}(\zeta_3)
Fixed field: Q(ζ7)+\Q(\zeta_{7})^+

Values on generators

33 → e(23)e\left(\frac{2}{3}\right)

Values

aa −1-11122334455
χ7(4,a) \chi_{ 7 }(4, a) 1111e(13)e\left(\frac{1}{3}\right)e(23)e\left(\frac{2}{3}\right)e(23)e\left(\frac{2}{3}\right)e(13)e\left(\frac{1}{3}\right)
sage: chi.jacobi_sum(n)
 
χ7(4,a)   \chi_{ 7 }(4,a) \; at   a=\;a = e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
τa(χ7(4,⋅))   \tau_{ a }( \chi_{ 7 }(4,·) )\; at   a=\;a = e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
J(χ7(4,⋅),χ7(n,⋅))   J(\chi_{ 7 }(4,·),\chi_{ 7 }(n,·)) \; for   n= \; n = e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
K(a,b,χ7(4,⋅))  K(a,b,\chi_{ 7 }(4,·)) \; at   a,b=\; a,b = e.g. 1,2