Properties

Label 1134.41
Modulus $1134$
Conductor $567$
Order $54$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1134, base_ring=CyclotomicField(54))
 
M = H._module
 
chi = DirichletCharacter(H, M([53,27]))
 
pari: [g,chi] = znchar(Mod(41,1134))
 

Basic properties

Modulus: \(1134\)
Conductor: \(567\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(54\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{567}(41,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1134.bq

\(\chi_{1134}(41,\cdot)\) \(\chi_{1134}(83,\cdot)\) \(\chi_{1134}(167,\cdot)\) \(\chi_{1134}(209,\cdot)\) \(\chi_{1134}(293,\cdot)\) \(\chi_{1134}(335,\cdot)\) \(\chi_{1134}(419,\cdot)\) \(\chi_{1134}(461,\cdot)\) \(\chi_{1134}(545,\cdot)\) \(\chi_{1134}(587,\cdot)\) \(\chi_{1134}(671,\cdot)\) \(\chi_{1134}(713,\cdot)\) \(\chi_{1134}(797,\cdot)\) \(\chi_{1134}(839,\cdot)\) \(\chi_{1134}(923,\cdot)\) \(\chi_{1134}(965,\cdot)\) \(\chi_{1134}(1049,\cdot)\) \(\chi_{1134}(1091,\cdot)\)

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: \(\Q(\zeta_{27})\)
Fixed field: Number field defined by a degree 54 polynomial

Values on generators

\((407,325)\) → \((e\left(\frac{53}{54}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 1134 }(41, a) \) \(1\)\(1\)\(e\left(\frac{2}{27}\right)\)\(e\left(\frac{41}{54}\right)\)\(e\left(\frac{19}{54}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{43}{54}\right)\)\(e\left(\frac{4}{27}\right)\)\(e\left(\frac{17}{54}\right)\)\(e\left(\frac{7}{54}\right)\)\(e\left(\frac{2}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1134 }(41,a) \;\) at \(\;a = \) e.g. 2