Properties

Label 3234.1013
Modulus $3234$
Conductor $147$
Order $42$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3234, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([21,41,0]))
 
pari: [g,chi] = znchar(Mod(1013,3234))
 

Basic properties

Modulus: \(3234\)
Conductor: \(147\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{147}(131,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3234.bw

\(\chi_{3234}(89,\cdot)\) \(\chi_{3234}(353,\cdot)\) \(\chi_{3234}(551,\cdot)\) \(\chi_{3234}(1013,\cdot)\) \(\chi_{3234}(1277,\cdot)\) \(\chi_{3234}(1475,\cdot)\) \(\chi_{3234}(1739,\cdot)\) \(\chi_{3234}(1937,\cdot)\) \(\chi_{3234}(2201,\cdot)\) \(\chi_{3234}(2399,\cdot)\) \(\chi_{3234}(2663,\cdot)\) \(\chi_{3234}(3125,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: \(\Q(\zeta_{147})^+\)

Values on generators

\((1079,199,2059)\) → \((-1,e\left(\frac{41}{42}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 3234 }(1013, a) \) \(1\)\(1\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{25}{42}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{1}{7}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3234 }(1013,a) \;\) at \(\;a = \) e.g. 2