Properties

Label 136.15
Modulus $136$
Conductor $68$
Order $8$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(136, base_ring=CyclotomicField(8))
 
M = H._module
 
chi = DirichletCharacter(H, M([4,0,3]))
 
pari: [g,chi] = znchar(Mod(15,136))
 

Basic properties

Modulus: \(136\)
Conductor: \(68\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(8\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{68}(15,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 136.m

\(\chi_{136}(15,\cdot)\) \(\chi_{136}(87,\cdot)\) \(\chi_{136}(111,\cdot)\) \(\chi_{136}(127,\cdot)\)

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: \(\Q(\zeta_{8})\)
Fixed field: 8.0.105046700288.1

Values on generators

\((103,69,105)\) → \((-1,1,e\left(\frac{3}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(19\)\(21\)\(23\)
\( \chi_{ 136 }(15, a) \) \(-1\)\(1\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{5}{8}\right)\)\(-i\)\(e\left(\frac{1}{8}\right)\)\(-1\)\(-i\)\(-i\)\(-1\)\(e\left(\frac{1}{8}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 136 }(15,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 136 }(15,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 136 }(15,·),\chi_{ 136 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 136 }(15,·)) \;\) at \(\; a,b = \) e.g. 1,2