Properties

Label 6027.1601
Modulus $6027$
Conductor $6027$
Order $420$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6027, base_ring=CyclotomicField(420))
 
M = H._module
 
chi = DirichletCharacter(H, M([210,410,273]))
 
pari: [g,chi] = znchar(Mod(1601,6027))
 

Basic properties

Modulus: \(6027\)
Conductor: \(6027\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(420\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6027.er

\(\chi_{6027}(5,\cdot)\) \(\chi_{6027}(131,\cdot)\) \(\chi_{6027}(143,\cdot)\) \(\chi_{6027}(185,\cdot)\) \(\chi_{6027}(248,\cdot)\) \(\chi_{6027}(320,\cdot)\) \(\chi_{6027}(446,\cdot)\) \(\chi_{6027}(500,\cdot)\) \(\chi_{6027}(572,\cdot)\) \(\chi_{6027}(635,\cdot)\) \(\chi_{6027}(677,\cdot)\) \(\chi_{6027}(689,\cdot)\) \(\chi_{6027}(740,\cdot)\) \(\chi_{6027}(866,\cdot)\) \(\chi_{6027}(941,\cdot)\) \(\chi_{6027}(992,\cdot)\) \(\chi_{6027}(1004,\cdot)\) \(\chi_{6027}(1046,\cdot)\) \(\chi_{6027}(1181,\cdot)\) \(\chi_{6027}(1235,\cdot)\) \(\chi_{6027}(1307,\cdot)\) \(\chi_{6027}(1361,\cdot)\) \(\chi_{6027}(1433,\cdot)\) \(\chi_{6027}(1496,\cdot)\) \(\chi_{6027}(1601,\cdot)\) \(\chi_{6027}(1676,\cdot)\) \(\chi_{6027}(1727,\cdot)\) \(\chi_{6027}(1802,\cdot)\) \(\chi_{6027}(1853,\cdot)\) \(\chi_{6027}(1865,\cdot)\) ...

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: $\Q(\zeta_{420})$
Fixed field: Number field defined by a degree 420 polynomial (not computed)

Values on generators

\((4019,493,2794)\) → \((-1,e\left(\frac{41}{42}\right),e\left(\frac{13}{20}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 6027 }(1601, a) \) \(1\)\(1\)\(e\left(\frac{82}{105}\right)\)\(e\left(\frac{59}{105}\right)\)\(e\left(\frac{23}{210}\right)\)\(e\left(\frac{12}{35}\right)\)\(e\left(\frac{187}{210}\right)\)\(e\left(\frac{209}{420}\right)\)\(e\left(\frac{51}{140}\right)\)\(e\left(\frac{13}{105}\right)\)\(e\left(\frac{149}{420}\right)\)\(e\left(\frac{1}{60}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6027 }(1601,a) \;\) at \(\;a = \) e.g. 2