Properties

Label 137.77
Modulus $137$
Conductor $137$
Order $34$
Real no
Primitive yes
Minimal yes
Parity even

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(77,137))
 

Basic properties

Modulus: \(137\)
Conductor: \(137\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(34\)
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 137.f

\(\chi_{137}(4,\cdot)\) \(\chi_{137}(14,\cdot)\) \(\chi_{137}(15,\cdot)\) \(\chi_{137}(18,\cdot)\) \(\chi_{137}(22,\cdot)\) \(\chi_{137}(49,\cdot)\) \(\chi_{137}(63,\cdot)\) \(\chi_{137}(64,\cdot)\) \(\chi_{137}(65,\cdot)\) \(\chi_{137}(77,\cdot)\) \(\chi_{137}(78,\cdot)\) \(\chi_{137}(81,\cdot)\) \(\chi_{137}(87,\cdot)\) \(\chi_{137}(99,\cdot)\) \(\chi_{137}(103,\cdot)\) \(\chi_{137}(121,\cdot)\)

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

\(3\) → \(e\left(\frac{7}{34}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{1}{17}\right)\)\(e\left(\frac{7}{34}\right)\)\(e\left(\frac{2}{17}\right)\)\(e\left(\frac{15}{34}\right)\)\(e\left(\frac{9}{34}\right)\)\(e\left(\frac{11}{17}\right)\)\(e\left(\frac{3}{17}\right)\)\(e\left(\frac{7}{17}\right)\)\(-1\)\(e\left(\frac{2}{17}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{17})\)
Fixed field: 34.34.32492169951483601485711825975325195410331229243708069666036144501372297.1

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 137 }(77,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{137}(77,\cdot)) = \sum_{r\in \Z/137\Z} \chi_{137}(77,r) e\left(\frac{2r}{137}\right) = 0.7528750012+11.6804614306i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 137 }(77,·),\chi_{ 137 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{137}(77,\cdot),\chi_{137}(1,\cdot)) = \sum_{r\in \Z/137\Z} \chi_{137}(77,r) \chi_{137}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 137 }(77,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{137}(77,·)) = \sum_{r \in \Z/137\Z} \chi_{137}(77,r) e\left(\frac{1 r + 2 r^{-1}}{137}\right) = 10.512843493+1.9651910346i \)