Properties

Label 6030.247
Modulus $6030$
Conductor $3015$
Order $132$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6030, base_ring=CyclotomicField(132))
 
M = H._module
 
chi = DirichletCharacter(H, M([44,33,58]))
 
pari: [g,chi] = znchar(Mod(247,6030))
 

Basic properties

Modulus: \(6030\)
Conductor: \(3015\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(132\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{3015}(247,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6030.ef

\(\chi_{6030}(247,\cdot)\) \(\chi_{6030}(727,\cdot)\) \(\chi_{6030}(733,\cdot)\) \(\chi_{6030}(817,\cdot)\) \(\chi_{6030}(1033,\cdot)\) \(\chi_{6030}(1123,\cdot)\) \(\chi_{6030}(1237,\cdot)\) \(\chi_{6030}(1267,\cdot)\) \(\chi_{6030}(1453,\cdot)\) \(\chi_{6030}(1687,\cdot)\) \(\chi_{6030}(1933,\cdot)\) \(\chi_{6030}(1987,\cdot)\) \(\chi_{6030}(2023,\cdot)\) \(\chi_{6030}(2347,\cdot)\) \(\chi_{6030}(2443,\cdot)\) \(\chi_{6030}(2473,\cdot)\) \(\chi_{6030}(2527,\cdot)\) \(\chi_{6030}(2587,\cdot)\) \(\chi_{6030}(2767,\cdot)\) \(\chi_{6030}(2797,\cdot)\) \(\chi_{6030}(2893,\cdot)\) \(\chi_{6030}(3193,\cdot)\) \(\chi_{6030}(3553,\cdot)\) \(\chi_{6030}(3733,\cdot)\) \(\chi_{6030}(3793,\cdot)\) \(\chi_{6030}(3973,\cdot)\) \(\chi_{6030}(4003,\cdot)\) \(\chi_{6030}(4027,\cdot)\) \(\chi_{6030}(4387,\cdot)\) \(\chi_{6030}(4507,\cdot)\) ...

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: $\Q(\zeta_{132})$
Fixed field: Number field defined by a degree 132 polynomial (not computed)

Values on generators

\((4691,1207,3151)\) → \((e\left(\frac{1}{3}\right),i,e\left(\frac{29}{66}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 6030 }(247, a) \) \(1\)\(1\)\(e\left(\frac{91}{132}\right)\)\(e\left(\frac{17}{66}\right)\)\(e\left(\frac{101}{132}\right)\)\(e\left(\frac{49}{132}\right)\)\(e\left(\frac{59}{66}\right)\)\(e\left(\frac{95}{132}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{21}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6030 }(247,a) \;\) at \(\;a = \) e.g. 2