Properties

Label 8041.897
Modulus $8041$
Conductor $8041$
Order $60$
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(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([54,15,10]))
 
pari: [g,chi] = znchar(Mod(897,8041))
 

Basic properties

Modulus: \(8041\)
Conductor: \(8041\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
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.dn

\(\chi_{8041}(123,\cdot)\) \(\chi_{8041}(480,\cdot)\) \(\chi_{8041}(523,\cdot)\) \(\chi_{8041}(854,\cdot)\) \(\chi_{8041}(897,\cdot)\) \(\chi_{8041}(1942,\cdot)\) \(\chi_{8041}(2316,\cdot)\) \(\chi_{8041}(5640,\cdot)\) \(\chi_{8041}(6014,\cdot)\) \(\chi_{8041}(6371,\cdot)\) \(\chi_{8041}(6745,\cdot)\) \(\chi_{8041}(7059,\cdot)\) \(\chi_{8041}(7102,\cdot)\) \(\chi_{8041}(7433,\cdot)\) \(\chi_{8041}(7476,\cdot)\) \(\chi_{8041}(7790,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((6580,2366,562)\) → \((e\left(\frac{9}{10}\right),i,e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 8041 }(897, a) \) \(1\)\(1\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{37}{60}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{1}{60}\right)\)\(e\left(\frac{31}{60}\right)\)\(e\left(\frac{53}{60}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{5}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8041 }(897,a) \;\) at \(\;a = \) e.g. 2