Properties

Label 6030.181
Modulus $6030$
Conductor $67$
Order $33$
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([0,0,50]))
 
pari: [g,chi] = znchar(Mod(181,6030))
 

Basic properties

Modulus: \(6030\)
Conductor: \(67\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(33\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{67}(47,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6030.ct

\(\chi_{6030}(181,\cdot)\) \(\chi_{6030}(361,\cdot)\) \(\chi_{6030}(451,\cdot)\) \(\chi_{6030}(1261,\cdot)\) \(\chi_{6030}(1711,\cdot)\) \(\chi_{6030}(2161,\cdot)\) \(\chi_{6030}(2431,\cdot)\) \(\chi_{6030}(2611,\cdot)\) \(\chi_{6030}(2701,\cdot)\) \(\chi_{6030}(2971,\cdot)\) \(\chi_{6030}(3421,\cdot)\) \(\chi_{6030}(3691,\cdot)\) \(\chi_{6030}(4141,\cdot)\) \(\chi_{6030}(4231,\cdot)\) \(\chi_{6030}(4321,\cdot)\) \(\chi_{6030}(4411,\cdot)\) \(\chi_{6030}(4591,\cdot)\) \(\chi_{6030}(4951,\cdot)\) \(\chi_{6030}(5041,\cdot)\) \(\chi_{6030}(5131,\cdot)\)

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 33 polynomial

Values on generators

\((4691,1207,3151)\) → \((1,1,e\left(\frac{25}{33}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 6030 }(181, a) \) \(1\)\(1\)\(e\left(\frac{14}{33}\right)\)\(e\left(\frac{23}{33}\right)\)\(e\left(\frac{13}{33}\right)\)\(e\left(\frac{16}{33}\right)\)\(e\left(\frac{19}{33}\right)\)\(e\left(\frac{7}{33}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{20}{33}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{33}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6030 }(181,a) \;\) at \(\;a = \) e.g. 2