Properties

Label 137.41
Modulus $137$
Conductor $137$
Order $8$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(137)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([7]))
 
pari: [g,chi] = znchar(Mod(41,137))
 

Basic properties

Modulus: \(137\)
Conductor: \(137\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(8\)
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 137.d

\(\chi_{137}(10,\cdot)\) \(\chi_{137}(41,\cdot)\) \(\chi_{137}(96,\cdot)\) \(\chi_{137}(127,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Values on generators

\(3\) → \(e\left(\frac{7}{8}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(-1\)\(1\)\(-i\)\(e\left(\frac{7}{8}\right)\)\(-1\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{5}{8}\right)\)\(-i\)\(i\)\(-i\)\(e\left(\frac{3}{8}\right)\)\(-i\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{8})\)
Fixed field: 8.0.905824306333433.1

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 137 }(41,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{137}(41,\cdot)) = \sum_{r\in \Z/137\Z} \chi_{137}(41,r) e\left(\frac{2r}{137}\right) = 6.3024828215+-9.8629970235i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 137 }(41,·),\chi_{ 137 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{137}(41,\cdot),\chi_{137}(1,\cdot)) = \sum_{r\in \Z/137\Z} \chi_{137}(41,r) \chi_{137}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 137 }(41,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{137}(41,·)) = \sum_{r \in \Z/137\Z} \chi_{137}(41,r) e\left(\frac{1 r + 2 r^{-1}}{137}\right) = 4.4589429105+4.4589429105i \)