Properties

Label 8041.5125
Modulus $8041$
Conductor $8041$
Order $56$
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(56))
 
M = H._module
 
chi = DirichletCharacter(H, M([28,35,52]))
 
pari: [g,chi] = znchar(Mod(5125,8041))
 

Basic properties

Modulus: \(8041\)
Conductor: \(8041\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(56\)
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.dh

\(\chi_{8041}(32,\cdot)\) \(\chi_{8041}(604,\cdot)\) \(\chi_{8041}(978,\cdot)\) \(\chi_{8041}(1011,\cdot)\) \(\chi_{8041}(1341,\cdot)\) \(\chi_{8041}(1759,\cdot)\) \(\chi_{8041}(1957,\cdot)\) \(\chi_{8041}(2287,\cdot)\) \(\chi_{8041}(2650,\cdot)\) \(\chi_{8041}(2705,\cdot)\) \(\chi_{8041}(3442,\cdot)\) \(\chi_{8041}(3596,\cdot)\) \(\chi_{8041}(3816,\cdot)\) \(\chi_{8041}(4388,\cdot)\) \(\chi_{8041}(4762,\cdot)\) \(\chi_{8041}(5125,\cdot)\) \(\chi_{8041}(5268,\cdot)\) \(\chi_{8041}(6016,\cdot)\) \(\chi_{8041}(6071,\cdot)\) \(\chi_{8041}(6214,\cdot)\) \(\chi_{8041}(6434,\cdot)\) \(\chi_{8041}(6962,\cdot)\) \(\chi_{8041}(7380,\cdot)\) \(\chi_{8041}(7699,\cdot)\)

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: $\Q(\zeta_{56})$
Fixed field: Number field defined by a degree 56 polynomial

Values on generators

\((6580,2366,562)\) → \((-1,e\left(\frac{5}{8}\right),e\left(\frac{13}{14}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 8041 }(5125, a) \) \(1\)\(1\)\(e\left(\frac{9}{28}\right)\)\(e\left(\frac{31}{56}\right)\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{19}{56}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{27}{28}\right)\)\(e\left(\frac{3}{28}\right)\)\(e\left(\frac{37}{56}\right)\)\(e\left(\frac{11}{56}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8041 }(5125,a) \;\) at \(\;a = \) e.g. 2