Properties

Label 6171.10
Modulus $6171$
Conductor $2057$
Order $176$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6171, base_ring=CyclotomicField(176))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,120,33]))
 
pari: [g,chi] = znchar(Mod(10,6171))
 

Basic properties

Modulus: \(6171\)
Conductor: \(2057\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(176\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2057}(10,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6171.cp

\(\chi_{6171}(10,\cdot)\) \(\chi_{6171}(109,\cdot)\) \(\chi_{6171}(142,\cdot)\) \(\chi_{6171}(175,\cdot)\) \(\chi_{6171}(439,\cdot)\) \(\chi_{6171}(505,\cdot)\) \(\chi_{6171}(538,\cdot)\) \(\chi_{6171}(571,\cdot)\) \(\chi_{6171}(670,\cdot)\) \(\chi_{6171}(703,\cdot)\) \(\chi_{6171}(736,\cdot)\) \(\chi_{6171}(802,\cdot)\) \(\chi_{6171}(1000,\cdot)\) \(\chi_{6171}(1066,\cdot)\) \(\chi_{6171}(1099,\cdot)\) \(\chi_{6171}(1132,\cdot)\) \(\chi_{6171}(1231,\cdot)\) \(\chi_{6171}(1264,\cdot)\) \(\chi_{6171}(1297,\cdot)\) \(\chi_{6171}(1363,\cdot)\) \(\chi_{6171}(1561,\cdot)\) \(\chi_{6171}(1627,\cdot)\) \(\chi_{6171}(1660,\cdot)\) \(\chi_{6171}(1792,\cdot)\) \(\chi_{6171}(1825,\cdot)\) \(\chi_{6171}(1858,\cdot)\) \(\chi_{6171}(1924,\cdot)\) \(\chi_{6171}(2122,\cdot)\) \(\chi_{6171}(2188,\cdot)\) \(\chi_{6171}(2221,\cdot)\) ...

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: $\Q(\zeta_{176})$
Fixed field: Number field defined by a degree 176 polynomial (not computed)

Values on generators

\((4115,970,2179)\) → \((1,e\left(\frac{15}{22}\right),e\left(\frac{3}{16}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(13\)\(14\)\(16\)\(19\)
\( \chi_{ 6171 }(10, a) \) \(1\)\(1\)\(e\left(\frac{27}{88}\right)\)\(e\left(\frac{27}{44}\right)\)\(e\left(\frac{69}{176}\right)\)\(e\left(\frac{147}{176}\right)\)\(e\left(\frac{81}{88}\right)\)\(e\left(\frac{123}{176}\right)\)\(e\left(\frac{27}{44}\right)\)\(e\left(\frac{25}{176}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{19}{88}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6171 }(10,a) \;\) at \(\;a = \) e.g. 2