Properties

Label 111.34
Modulus $111$
Conductor $37$
Order $9$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(111, base_ring=CyclotomicField(18))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,4]))
 
pari: [g,chi] = znchar(Mod(34,111))
 

Basic properties

Modulus: \(111\)
Conductor: \(37\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(9\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{37}(34,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 111.k

\(\chi_{111}(7,\cdot)\) \(\chi_{111}(16,\cdot)\) \(\chi_{111}(34,\cdot)\) \(\chi_{111}(46,\cdot)\) \(\chi_{111}(49,\cdot)\) \(\chi_{111}(70,\cdot)\)

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: \(\Q(\zeta_{9})\)
Fixed field: 9.9.3512479453921.1

Values on generators

\((38,76)\) → \((1,e\left(\frac{2}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 111 }(34, a) \) \(1\)\(1\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{8}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 111 }(34,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 111 }(34,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 111 }(34,·),\chi_{ 111 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 111 }(34,·)) \;\) at \(\; a,b = \) e.g. 1,2