Properties

Label 6027.2710
Modulus $6027$
Conductor $2009$
Order $70$
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(70))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,50,21]))
 
pari: [g,chi] = znchar(Mod(2710,6027))
 

Basic properties

Modulus: \(6027\)
Conductor: \(2009\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(70\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2009}(701,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6027.dn

\(\chi_{6027}(64,\cdot)\) \(\chi_{6027}(127,\cdot)\) \(\chi_{6027}(400,\cdot)\) \(\chi_{6027}(925,\cdot)\) \(\chi_{6027}(988,\cdot)\) \(\chi_{6027}(1009,\cdot)\) \(\chi_{6027}(1261,\cdot)\) \(\chi_{6027}(1786,\cdot)\) \(\chi_{6027}(1849,\cdot)\) \(\chi_{6027}(1870,\cdot)\) \(\chi_{6027}(2122,\cdot)\) \(\chi_{6027}(2710,\cdot)\) \(\chi_{6027}(2731,\cdot)\) \(\chi_{6027}(2983,\cdot)\) \(\chi_{6027}(3508,\cdot)\) \(\chi_{6027}(3571,\cdot)\) \(\chi_{6027}(3592,\cdot)\) \(\chi_{6027}(3844,\cdot)\) \(\chi_{6027}(4369,\cdot)\) \(\chi_{6027}(4432,\cdot)\) \(\chi_{6027}(4453,\cdot)\) \(\chi_{6027}(5230,\cdot)\) \(\chi_{6027}(5314,\cdot)\) \(\chi_{6027}(5566,\cdot)\)

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: $\Q(\zeta_{35})$
Fixed field: Number field defined by a degree 70 polynomial

Values on generators

\((4019,493,2794)\) → \((1,e\left(\frac{5}{7}\right),e\left(\frac{3}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 6027 }(2710, a) \) \(1\)\(1\)\(e\left(\frac{13}{35}\right)\)\(e\left(\frac{26}{35}\right)\)\(e\left(\frac{11}{35}\right)\)\(e\left(\frac{4}{35}\right)\)\(e\left(\frac{24}{35}\right)\)\(e\left(\frac{33}{70}\right)\)\(e\left(\frac{61}{70}\right)\)\(e\left(\frac{17}{35}\right)\)\(e\left(\frac{53}{70}\right)\)\(e\left(\frac{7}{10}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6027 }(2710,a) \;\) at \(\;a = \) e.g. 2