Properties

Label 8041.2817
Modulus $8041$
Conductor $731$
Order $112$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8041, base_ring=CyclotomicField(112))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,91,40]))
 
pari: [g,chi] = znchar(Mod(2817,8041))
 

Basic properties

Modulus: \(8041\)
Conductor: \(731\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(112\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{731}(624,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8041.eh

\(\chi_{8041}(45,\cdot)\) \(\chi_{8041}(199,\cdot)\) \(\chi_{8041}(309,\cdot)\) \(\chi_{8041}(419,\cdot)\) \(\chi_{8041}(452,\cdot)\) \(\chi_{8041}(925,\cdot)\) \(\chi_{8041}(991,\cdot)\) \(\chi_{8041}(1145,\cdot)\) \(\chi_{8041}(1200,\cdot)\) \(\chi_{8041}(1255,\cdot)\) \(\chi_{8041}(1365,\cdot)\) \(\chi_{8041}(1618,\cdot)\) \(\chi_{8041}(1673,\cdot)\) \(\chi_{8041}(1728,\cdot)\) \(\chi_{8041}(2564,\cdot)\) \(\chi_{8041}(2674,\cdot)\) \(\chi_{8041}(2817,\cdot)\) \(\chi_{8041}(2883,\cdot)\) \(\chi_{8041}(3037,\cdot)\) \(\chi_{8041}(3257,\cdot)\) \(\chi_{8041}(3356,\cdot)\) \(\chi_{8041}(3565,\cdot)\) \(\chi_{8041}(3730,\cdot)\) \(\chi_{8041}(3763,\cdot)\) \(\chi_{8041}(3983,\cdot)\) \(\chi_{8041}(4236,\cdot)\) \(\chi_{8041}(4511,\cdot)\) \(\chi_{8041}(4566,\cdot)\) \(\chi_{8041}(4984,\cdot)\) \(\chi_{8041}(5039,\cdot)\) ...

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: $\Q(\zeta_{112})$
Fixed field: Number field defined by a degree 112 polynomial (not computed)

Values on generators

\((6580,2366,562)\) → \((1,e\left(\frac{13}{16}\right),e\left(\frac{5}{14}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 8041 }(2817, a) \) \(1\)\(1\)\(e\left(\frac{1}{56}\right)\)\(e\left(\frac{19}{112}\right)\)\(e\left(\frac{1}{28}\right)\)\(e\left(\frac{111}{112}\right)\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{3}{56}\right)\)\(e\left(\frac{19}{56}\right)\)\(e\left(\frac{1}{112}\right)\)\(e\left(\frac{23}{112}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8041 }(2817,a) \;\) at \(\;a = \) e.g. 2