Properties

Label 137.119
Modulus $137$
Conductor $137$
Order $17$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(137, base_ring=CyclotomicField(34))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([20]))
 
pari: [g,chi] = znchar(Mod(119,137))
 

Basic properties

Modulus: \(137\)
Conductor: \(137\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(17\)
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.e

\(\chi_{137}(16,\cdot)\) \(\chi_{137}(34,\cdot)\) \(\chi_{137}(38,\cdot)\) \(\chi_{137}(50,\cdot)\) \(\chi_{137}(56,\cdot)\) \(\chi_{137}(59,\cdot)\) \(\chi_{137}(60,\cdot)\) \(\chi_{137}(72,\cdot)\) \(\chi_{137}(73,\cdot)\) \(\chi_{137}(74,\cdot)\) \(\chi_{137}(88,\cdot)\) \(\chi_{137}(115,\cdot)\) \(\chi_{137}(119,\cdot)\) \(\chi_{137}(122,\cdot)\) \(\chi_{137}(123,\cdot)\) \(\chi_{137}(133,\cdot)\)

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

Related number fields

Field of values: \(\Q(\zeta_{17})\)
Fixed field: 17.17.15400296222263289476715621650663041.1

Values on generators

\(3\) → \(e\left(\frac{10}{17}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{15}{17}\right)\)\(e\left(\frac{10}{17}\right)\)\(e\left(\frac{13}{17}\right)\)\(e\left(\frac{2}{17}\right)\)\(e\left(\frac{8}{17}\right)\)\(e\left(\frac{12}{17}\right)\)\(e\left(\frac{11}{17}\right)\)\(e\left(\frac{3}{17}\right)\)\(1\)\(e\left(\frac{13}{17}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 137 }(119,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{137}(119,\cdot)) = \sum_{r\in \Z/137\Z} \chi_{137}(119,r) e\left(\frac{2r}{137}\right) = 1.875636227+-11.5534405587i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 137 }(119,·),\chi_{ 137 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{137}(119,\cdot),\chi_{137}(1,\cdot)) = \sum_{r\in \Z/137\Z} \chi_{137}(119,r) \chi_{137}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 137 }(119,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{137}(119,·)) = \sum_{r \in \Z/137\Z} \chi_{137}(119,r) e\left(\frac{1 r + 2 r^{-1}}{137}\right) = -1.4803093903+0.573474913i \)