Properties

Label 6171.16
Modulus $6171$
Conductor $2057$
Order $110$
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(110))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,4,55]))
 
pari: [g,chi] = znchar(Mod(16,6171))
 

Basic properties

Modulus: \(6171\)
Conductor: \(2057\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(110\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2057}(16,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6171.ck

\(\chi_{6171}(16,\cdot)\) \(\chi_{6171}(169,\cdot)\) \(\chi_{6171}(322,\cdot)\) \(\chi_{6171}(526,\cdot)\) \(\chi_{6171}(577,\cdot)\) \(\chi_{6171}(730,\cdot)\) \(\chi_{6171}(883,\cdot)\) \(\chi_{6171}(1087,\cdot)\) \(\chi_{6171}(1138,\cdot)\) \(\chi_{6171}(1444,\cdot)\) \(\chi_{6171}(1648,\cdot)\) \(\chi_{6171}(1699,\cdot)\) \(\chi_{6171}(1852,\cdot)\) \(\chi_{6171}(2005,\cdot)\) \(\chi_{6171}(2209,\cdot)\) \(\chi_{6171}(2260,\cdot)\) \(\chi_{6171}(2413,\cdot)\) \(\chi_{6171}(2566,\cdot)\) \(\chi_{6171}(2770,\cdot)\) \(\chi_{6171}(2821,\cdot)\) \(\chi_{6171}(2974,\cdot)\) \(\chi_{6171}(3127,\cdot)\) \(\chi_{6171}(3331,\cdot)\) \(\chi_{6171}(3382,\cdot)\) \(\chi_{6171}(3535,\cdot)\) \(\chi_{6171}(3688,\cdot)\) \(\chi_{6171}(3892,\cdot)\) \(\chi_{6171}(3943,\cdot)\) \(\chi_{6171}(4096,\cdot)\) \(\chi_{6171}(4249,\cdot)\) ...

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: $\Q(\zeta_{55})$
Fixed field: Number field defined by a degree 110 polynomial (not computed)

Values on generators

\((4115,970,2179)\) → \((1,e\left(\frac{2}{55}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(13\)\(14\)\(16\)\(19\)
\( \chi_{ 6171 }(16, a) \) \(1\)\(1\)\(e\left(\frac{2}{55}\right)\)\(e\left(\frac{4}{55}\right)\)\(e\left(\frac{21}{110}\right)\)\(e\left(\frac{83}{110}\right)\)\(e\left(\frac{6}{55}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{37}{55}\right)\)\(e\left(\frac{87}{110}\right)\)\(e\left(\frac{8}{55}\right)\)\(e\left(\frac{1}{55}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6171 }(16,a) \;\) at \(\;a = \) e.g. 2