Properties

Label 234.227
Modulus $234$
Conductor $117$
Order $12$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(234, base_ring=CyclotomicField(12))
 
M = H._module
 
chi = DirichletCharacter(H, M([2,5]))
 
pari: [g,chi] = znchar(Mod(227,234))
 

Basic properties

Modulus: \(234\)
Conductor: \(117\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{117}(110,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 234.z

\(\chi_{234}(41,\cdot)\) \(\chi_{234}(137,\cdot)\) \(\chi_{234}(167,\cdot)\) \(\chi_{234}(227,\cdot)\)

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: \(\Q(\zeta_{12})\)
Fixed field: 12.12.694319656224247224093.1

Values on generators

\((209,145)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{5}{12}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 234 }(227, a) \) \(1\)\(1\)\(e\left(\frac{7}{12}\right)\)\(i\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{12}\right)\)\(1\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{5}{6}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 234 }(227,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 234 }(227,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 234 }(227,·),\chi_{ 234 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 234 }(227,·)) \;\) at \(\; a,b = \) e.g. 1,2