Properties

Label 6027.622
Modulus $6027$
Conductor $2009$
Order $280$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6027, base_ring=CyclotomicField(280))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,60,273]))
 
pari: [g,chi] = znchar(Mod(622,6027))
 

Basic properties

Modulus: \(6027\)
Conductor: \(2009\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(280\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2009}(622,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6027.em

\(\chi_{6027}(13,\cdot)\) \(\chi_{6027}(34,\cdot)\) \(\chi_{6027}(76,\cdot)\) \(\chi_{6027}(181,\cdot)\) \(\chi_{6027}(265,\cdot)\) \(\chi_{6027}(475,\cdot)\) \(\chi_{6027}(559,\cdot)\) \(\chi_{6027}(580,\cdot)\) \(\chi_{6027}(622,\cdot)\) \(\chi_{6027}(643,\cdot)\) \(\chi_{6027}(727,\cdot)\) \(\chi_{6027}(790,\cdot)\) \(\chi_{6027}(874,\cdot)\) \(\chi_{6027}(895,\cdot)\) \(\chi_{6027}(937,\cdot)\) \(\chi_{6027}(958,\cdot)\) \(\chi_{6027}(1042,\cdot)\) \(\chi_{6027}(1252,\cdot)\) \(\chi_{6027}(1336,\cdot)\) \(\chi_{6027}(1441,\cdot)\) \(\chi_{6027}(1483,\cdot)\) \(\chi_{6027}(1504,\cdot)\) \(\chi_{6027}(1546,\cdot)\) \(\chi_{6027}(1588,\cdot)\) \(\chi_{6027}(1651,\cdot)\) \(\chi_{6027}(1693,\cdot)\) \(\chi_{6027}(1735,\cdot)\) \(\chi_{6027}(1756,\cdot)\) \(\chi_{6027}(1798,\cdot)\) \(\chi_{6027}(1819,\cdot)\) ...

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: $\Q(\zeta_{280})$
Fixed field: Number field defined by a degree 280 polynomial (not computed)

Values on generators

\((4019,493,2794)\) → \((1,e\left(\frac{3}{14}\right),e\left(\frac{39}{40}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 6027 }(622, a) \) \(1\)\(1\)\(e\left(\frac{129}{140}\right)\)\(e\left(\frac{59}{70}\right)\)\(e\left(\frac{93}{140}\right)\)\(e\left(\frac{107}{140}\right)\)\(e\left(\frac{41}{70}\right)\)\(e\left(\frac{139}{280}\right)\)\(e\left(\frac{83}{280}\right)\)\(e\left(\frac{24}{35}\right)\)\(e\left(\frac{149}{280}\right)\)\(e\left(\frac{11}{40}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6027 }(622,a) \;\) at \(\;a = \) e.g. 2