Properties

Label 137.23
Modulus $137$
Conductor $137$
Order $136$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(137, base_ring=CyclotomicField(136))
 
M = H._module
 
chi = DirichletCharacter(H, M([125]))
 
pari: [g,chi] = znchar(Mod(23,137))
 

Basic properties

Modulus: \(137\)
Conductor: \(137\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(136\)
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.h

\(\chi_{137}(3,\cdot)\) \(\chi_{137}(5,\cdot)\) \(\chi_{137}(6,\cdot)\) \(\chi_{137}(12,\cdot)\) \(\chi_{137}(13,\cdot)\) \(\chi_{137}(20,\cdot)\) \(\chi_{137}(21,\cdot)\) \(\chi_{137}(23,\cdot)\) \(\chi_{137}(24,\cdot)\) \(\chi_{137}(26,\cdot)\) \(\chi_{137}(27,\cdot)\) \(\chi_{137}(29,\cdot)\) \(\chi_{137}(31,\cdot)\) \(\chi_{137}(33,\cdot)\) \(\chi_{137}(35,\cdot)\) \(\chi_{137}(40,\cdot)\) \(\chi_{137}(42,\cdot)\) \(\chi_{137}(43,\cdot)\) \(\chi_{137}(45,\cdot)\) \(\chi_{137}(46,\cdot)\) \(\chi_{137}(47,\cdot)\) \(\chi_{137}(48,\cdot)\) \(\chi_{137}(51,\cdot)\) \(\chi_{137}(52,\cdot)\) \(\chi_{137}(53,\cdot)\) \(\chi_{137}(54,\cdot)\) \(\chi_{137}(55,\cdot)\) \(\chi_{137}(57,\cdot)\) \(\chi_{137}(58,\cdot)\) \(\chi_{137}(62,\cdot)\) ...

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: $\Q(\zeta_{136})$
Fixed field: Number field defined by a degree 136 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{125}{136}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 137 }(23, a) \) \(-1\)\(1\)\(e\left(\frac{13}{68}\right)\)\(e\left(\frac{125}{136}\right)\)\(e\left(\frac{13}{34}\right)\)\(e\left(\frac{127}{136}\right)\)\(e\left(\frac{15}{136}\right)\)\(e\left(\frac{41}{68}\right)\)\(e\left(\frac{39}{68}\right)\)\(e\left(\frac{57}{68}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{9}{68}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 137 }(23,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 137 }(23,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 137 }(23,·),\chi_{ 137 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 137 }(23,·)) \;\) at \(\; a,b = \) e.g. 1,2