Properties

Label 841.571
Modulus $841$
Conductor $29$
Order $7$
Real no
Primitive no
Minimal no
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(841, base_ring=CyclotomicField(14))
 
M = H._module
 
chi = DirichletCharacter(H, M([12]))
 
pari: [g,chi] = znchar(Mod(571,841))
 

Basic properties

Modulus: \(841\)
Conductor: \(29\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(7\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{29}(20,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 841.d

\(\chi_{841}(190,\cdot)\) \(\chi_{841}(571,\cdot)\) \(\chi_{841}(574,\cdot)\) \(\chi_{841}(605,\cdot)\) \(\chi_{841}(645,\cdot)\) \(\chi_{841}(778,\cdot)\)

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: \(\Q(\zeta_{7})\)
Fixed field: 7.7.594823321.1

Values on generators

\(2\) → \(e\left(\frac{6}{7}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 841 }(571, a) \) \(1\)\(1\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{3}{7}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 841 }(571,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 841 }(571,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 841 }(571,·),\chi_{ 841 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 841 }(571,·)) \;\) at \(\; a,b = \) e.g. 1,2