Properties

Label 8041.4058
Modulus $8041$
Conductor $8041$
Order $112$
Real no
Primitive yes
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([56,91,64]))
 
pari: [g,chi] = znchar(Mod(4058,8041))
 

Basic properties

Modulus: \(8041\)
Conductor: \(8041\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(112\)
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 8041.ei

\(\chi_{8041}(54,\cdot)\) \(\chi_{8041}(164,\cdot)\) \(\chi_{8041}(428,\cdot)\) \(\chi_{8041}(692,\cdot)\) \(\chi_{8041}(1000,\cdot)\) \(\chi_{8041}(1110,\cdot)\) \(\chi_{8041}(1374,\cdot)\) \(\chi_{8041}(1440,\cdot)\) \(\chi_{8041}(1473,\cdot)\) \(\chi_{8041}(1638,\cdot)\) \(\chi_{8041}(1693,\cdot)\) \(\chi_{8041}(1847,\cdot)\) \(\chi_{8041}(2111,\cdot)\) \(\chi_{8041}(2166,\cdot)\) \(\chi_{8041}(2386,\cdot)\) \(\chi_{8041}(2419,\cdot)\) \(\chi_{8041}(2793,\cdot)\) \(\chi_{8041}(2859,\cdot)\) \(\chi_{8041}(3002,\cdot)\) \(\chi_{8041}(3057,\cdot)\) \(\chi_{8041}(3475,\cdot)\) \(\chi_{8041}(3530,\cdot)\) \(\chi_{8041}(3805,\cdot)\) \(\chi_{8041}(4058,\cdot)\) \(\chi_{8041}(4278,\cdot)\) \(\chi_{8041}(4311,\cdot)\) \(\chi_{8041}(4476,\cdot)\) \(\chi_{8041}(4685,\cdot)\) \(\chi_{8041}(4784,\cdot)\) \(\chi_{8041}(5004,\cdot)\) ...

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

\((6580,2366,562)\) → \((-1,e\left(\frac{13}{16}\right),e\left(\frac{4}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 8041 }(4058, a) \) \(1\)\(1\)\(e\left(\frac{17}{56}\right)\)\(e\left(\frac{43}{112}\right)\)\(e\left(\frac{17}{28}\right)\)\(e\left(\frac{39}{112}\right)\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{51}{56}\right)\)\(e\left(\frac{43}{56}\right)\)\(e\left(\frac{73}{112}\right)\)\(e\left(\frac{111}{112}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8041 }(4058,a) \;\) at \(\;a = \) e.g. 2