Properties

Label 6030.349
Modulus $6030$
Conductor $3015$
Order $66$
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(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([44,33,24]))
 
pari: [g,chi] = znchar(Mod(349,6030))
 

Basic properties

Modulus: \(6030\)
Conductor: \(3015\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{3015}(349,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6030.dl

\(\chi_{6030}(349,\cdot)\) \(\chi_{6030}(679,\cdot)\) \(\chi_{6030}(799,\cdot)\) \(\chi_{6030}(1069,\cdot)\) \(\chi_{6030}(1399,\cdot)\) \(\chi_{6030}(1429,\cdot)\) \(\chi_{6030}(1489,\cdot)\) \(\chi_{6030}(1699,\cdot)\) \(\chi_{6030}(1849,\cdot)\) \(\chi_{6030}(2689,\cdot)\) \(\chi_{6030}(2839,\cdot)\) \(\chi_{6030}(3409,\cdot)\) \(\chi_{6030}(3499,\cdot)\) \(\chi_{6030}(3859,\cdot)\) \(\chi_{6030}(4369,\cdot)\) \(\chi_{6030}(4819,\cdot)\) \(\chi_{6030}(4849,\cdot)\) \(\chi_{6030}(5089,\cdot)\) \(\chi_{6030}(5449,\cdot)\) \(\chi_{6030}(5719,\cdot)\)

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: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 66 polynomial

Values on generators

\((4691,1207,3151)\) → \((e\left(\frac{2}{3}\right),-1,e\left(\frac{4}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 6030 }(349, a) \) \(1\)\(1\)\(e\left(\frac{35}{66}\right)\)\(e\left(\frac{4}{33}\right)\)\(e\left(\frac{49}{66}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{1}{66}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{14}{33}\right)\)\(-1\)\(e\left(\frac{20}{33}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6030 }(349,a) \;\) at \(\;a = \) e.g. 2