Properties

Label 137.129
Modulus $137$
Conductor $137$
Order $68$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(137, base_ring=CyclotomicField(68))
 
M = H._module
 
chi = DirichletCharacter(H, M([49]))
 
pari: [g,chi] = znchar(Mod(129,137))
 

Basic properties

Modulus: \(137\)
Conductor: \(137\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(68\)
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.g

\(\chi_{137}(2,\cdot)\) \(\chi_{137}(7,\cdot)\) \(\chi_{137}(8,\cdot)\) \(\chi_{137}(9,\cdot)\) \(\chi_{137}(11,\cdot)\) \(\chi_{137}(17,\cdot)\) \(\chi_{137}(19,\cdot)\) \(\chi_{137}(25,\cdot)\) \(\chi_{137}(28,\cdot)\) \(\chi_{137}(30,\cdot)\) \(\chi_{137}(32,\cdot)\) \(\chi_{137}(36,\cdot)\) \(\chi_{137}(39,\cdot)\) \(\chi_{137}(44,\cdot)\) \(\chi_{137}(61,\cdot)\) \(\chi_{137}(68,\cdot)\) \(\chi_{137}(69,\cdot)\) \(\chi_{137}(76,\cdot)\) \(\chi_{137}(93,\cdot)\) \(\chi_{137}(98,\cdot)\) \(\chi_{137}(101,\cdot)\) \(\chi_{137}(105,\cdot)\) \(\chi_{137}(107,\cdot)\) \(\chi_{137}(109,\cdot)\) \(\chi_{137}(112,\cdot)\) \(\chi_{137}(118,\cdot)\) \(\chi_{137}(120,\cdot)\) \(\chi_{137}(126,\cdot)\) \(\chi_{137}(128,\cdot)\) \(\chi_{137}(129,\cdot)\) ...

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: $\Q(\zeta_{68})$
Fixed field: Number field defined by a degree 68 polynomial

Values on generators

\(3\) → \(e\left(\frac{49}{68}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 137 }(129, a) \) \(1\)\(1\)\(e\left(\frac{7}{34}\right)\)\(e\left(\frac{49}{68}\right)\)\(e\left(\frac{7}{17}\right)\)\(e\left(\frac{3}{68}\right)\)\(e\left(\frac{63}{68}\right)\)\(e\left(\frac{9}{34}\right)\)\(e\left(\frac{21}{34}\right)\)\(e\left(\frac{15}{34}\right)\)\(i\)\(e\left(\frac{31}{34}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 137 }(129,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 137 }(129,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 137 }(129,·),\chi_{ 137 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 137 }(129,·)) \;\) at \(\; a,b = \) e.g. 1,2