Properties

Label 8041.4122
Modulus $8041$
Conductor $8041$
Order $120$
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(120))
 
M = H._module
 
chi = DirichletCharacter(H, M([36,75,20]))
 
pari: [g,chi] = znchar(Mod(4122,8041))
 

Basic properties

Modulus: \(8041\)
Conductor: \(8041\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(120\)
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.em

\(\chi_{8041}(338,\cdot)\) \(\chi_{8041}(695,\cdot)\) \(\chi_{8041}(910,\cdot)\) \(\chi_{8041}(1069,\cdot)\) \(\chi_{8041}(1284,\cdot)\) \(\chi_{8041}(1426,\cdot)\) \(\chi_{8041}(1641,\cdot)\) \(\chi_{8041}(1800,\cdot)\) \(\chi_{8041}(2015,\cdot)\) \(\chi_{8041}(2372,\cdot)\) \(\chi_{8041}(2746,\cdot)\) \(\chi_{8041}(2888,\cdot)\) \(\chi_{8041}(3262,\cdot)\) \(\chi_{8041}(3748,\cdot)\) \(\chi_{8041}(3834,\cdot)\) \(\chi_{8041}(4122,\cdot)\) \(\chi_{8041}(4208,\cdot)\) \(\chi_{8041}(4479,\cdot)\) \(\chi_{8041}(4694,\cdot)\) \(\chi_{8041}(4853,\cdot)\) \(\chi_{8041}(5068,\cdot)\) \(\chi_{8041}(5210,\cdot)\) \(\chi_{8041}(5425,\cdot)\) \(\chi_{8041}(5584,\cdot)\) \(\chi_{8041}(5799,\cdot)\) \(\chi_{8041}(6156,\cdot)\) \(\chi_{8041}(6530,\cdot)\) \(\chi_{8041}(6672,\cdot)\) \(\chi_{8041}(7046,\cdot)\) \(\chi_{8041}(7618,\cdot)\) ...

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: $\Q(\zeta_{120})$
Fixed field: Number field defined by a degree 120 polynomial (not computed)

Values on generators

\((6580,2366,562)\) → \((e\left(\frac{3}{10}\right),e\left(\frac{5}{8}\right),e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 8041 }(4122, a) \) \(1\)\(1\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{23}{120}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{59}{120}\right)\)\(e\left(\frac{89}{120}\right)\)\(e\left(\frac{97}{120}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{23}{60}\right)\)\(e\left(\frac{1}{24}\right)\)\(e\left(\frac{7}{24}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8041 }(4122,a) \;\) at \(\;a = \) e.g. 2