Properties

Label 959.29
Modulus $959$
Conductor $137$
Order $136$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(959, base_ring=CyclotomicField(136))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,91]))
 
pari: [g,chi] = znchar(Mod(29,959))
 

Basic properties

Modulus: \(959\)
Conductor: \(137\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(136\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{137}(29,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 959.ba

\(\chi_{959}(29,\cdot)\) \(\chi_{959}(43,\cdot)\) \(\chi_{959}(57,\cdot)\) \(\chi_{959}(71,\cdot)\) \(\chi_{959}(85,\cdot)\) \(\chi_{959}(92,\cdot)\) \(\chi_{959}(106,\cdot)\) \(\chi_{959}(113,\cdot)\) \(\chi_{959}(134,\cdot)\) \(\chi_{959}(183,\cdot)\) \(\chi_{959}(190,\cdot)\) \(\chi_{959}(204,\cdot)\) \(\chi_{959}(232,\cdot)\) \(\chi_{959}(239,\cdot)\) \(\chi_{959}(253,\cdot)\) \(\chi_{959}(295,\cdot)\) \(\chi_{959}(309,\cdot)\) \(\chi_{959}(316,\cdot)\) \(\chi_{959}(344,\cdot)\) \(\chi_{959}(358,\cdot)\) \(\chi_{959}(365,\cdot)\) \(\chi_{959}(414,\cdot)\) \(\chi_{959}(435,\cdot)\) \(\chi_{959}(442,\cdot)\) \(\chi_{959}(456,\cdot)\) \(\chi_{959}(463,\cdot)\) \(\chi_{959}(477,\cdot)\) \(\chi_{959}(491,\cdot)\) \(\chi_{959}(505,\cdot)\) \(\chi_{959}(519,\cdot)\) ...

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

\((549,414)\) → \((1,e\left(\frac{91}{136}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 959 }(29, a) \) \(-1\)\(1\)\(e\left(\frac{47}{68}\right)\)\(e\left(\frac{91}{136}\right)\)\(e\left(\frac{13}{34}\right)\)\(e\left(\frac{25}{136}\right)\)\(e\left(\frac{49}{136}\right)\)\(e\left(\frac{5}{68}\right)\)\(e\left(\frac{23}{68}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{43}{68}\right)\)\(e\left(\frac{7}{136}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 959 }(29,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 959 }(29,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 959 }(29,·),\chi_{ 959 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 959 }(29,·)) \;\) at \(\; a,b = \) e.g. 1,2