Properties

Label 6027.4049
Modulus $6027$
Conductor $861$
Order $30$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6027, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([15,5,21]))
 
pari: [g,chi] = znchar(Mod(4049,6027))
 

Basic properties

Modulus: \(6027\)
Conductor: \(861\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{861}(605,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6027.cj

\(\chi_{6027}(1097,\cdot)\) \(\chi_{6027}(1685,\cdot)\) \(\chi_{6027}(1991,\cdot)\) \(\chi_{6027}(2567,\cdot)\) \(\chi_{6027}(4049,\cdot)\) \(\chi_{6027}(4637,\cdot)\) \(\chi_{6027}(5066,\cdot)\) \(\chi_{6027}(5519,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: Number field defined by a degree 30 polynomial

Values on generators

\((4019,493,2794)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{7}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 6027 }(4049, a) \) \(1\)\(1\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{2}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6027 }(4049,a) \;\) at \(\;a = \) e.g. 2