Properties

Label 6027.1553
Modulus $6027$
Conductor $6027$
Order $140$
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(140))
 
M = H._module
 
chi = DirichletCharacter(H, M([70,30,7]))
 
pari: [g,chi] = znchar(Mod(1553,6027))
 

Basic properties

Modulus: \(6027\)
Conductor: \(6027\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(140\)
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.dz

\(\chi_{6027}(20,\cdot)\) \(\chi_{6027}(62,\cdot)\) \(\chi_{6027}(125,\cdot)\) \(\chi_{6027}(251,\cdot)\) \(\chi_{6027}(377,\cdot)\) \(\chi_{6027}(566,\cdot)\) \(\chi_{6027}(692,\cdot)\) \(\chi_{6027}(818,\cdot)\) \(\chi_{6027}(923,\cdot)\) \(\chi_{6027}(986,\cdot)\) \(\chi_{6027}(1112,\cdot)\) \(\chi_{6027}(1238,\cdot)\) \(\chi_{6027}(1427,\cdot)\) \(\chi_{6027}(1553,\cdot)\) \(\chi_{6027}(1679,\cdot)\) \(\chi_{6027}(1742,\cdot)\) \(\chi_{6027}(1784,\cdot)\) \(\chi_{6027}(1847,\cdot)\) \(\chi_{6027}(1973,\cdot)\) \(\chi_{6027}(2099,\cdot)\) \(\chi_{6027}(2288,\cdot)\) \(\chi_{6027}(2414,\cdot)\) \(\chi_{6027}(2540,\cdot)\) \(\chi_{6027}(2603,\cdot)\) \(\chi_{6027}(2708,\cdot)\) \(\chi_{6027}(2834,\cdot)\) \(\chi_{6027}(2960,\cdot)\) \(\chi_{6027}(3149,\cdot)\) \(\chi_{6027}(3275,\cdot)\) \(\chi_{6027}(3401,\cdot)\) ...

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: $\Q(\zeta_{140})$
Fixed field: Number field defined by a degree 140 polynomial (not computed)

Values on generators

\((4019,493,2794)\) → \((-1,e\left(\frac{3}{14}\right),e\left(\frac{1}{20}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 6027 }(1553, a) \) \(1\)\(1\)\(e\left(\frac{13}{35}\right)\)\(e\left(\frac{26}{35}\right)\)\(e\left(\frac{57}{70}\right)\)\(e\left(\frac{4}{35}\right)\)\(e\left(\frac{13}{70}\right)\)\(e\left(\frac{31}{140}\right)\)\(e\left(\frac{87}{140}\right)\)\(e\left(\frac{17}{35}\right)\)\(e\left(\frac{71}{140}\right)\)\(e\left(\frac{19}{20}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6027 }(1553,a) \;\) at \(\;a = \) e.g. 2