Properties

Label 8041.2323
Modulus $8041$
Conductor $187$
Order $80$
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(80))
 
M = H._module
 
chi = DirichletCharacter(H, M([8,35,0]))
 
pari: [g,chi] = znchar(Mod(2323,8041))
 

Basic properties

Modulus: \(8041\)
Conductor: \(187\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(80\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{187}(79,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8041.dz

\(\chi_{8041}(173,\cdot)\) \(\chi_{8041}(216,\cdot)\) \(\chi_{8041}(431,\cdot)\) \(\chi_{8041}(904,\cdot)\) \(\chi_{8041}(1119,\cdot)\) \(\chi_{8041}(1162,\cdot)\) \(\chi_{8041}(1592,\cdot)\) \(\chi_{8041}(1635,\cdot)\) \(\chi_{8041}(1678,\cdot)\) \(\chi_{8041}(1850,\cdot)\) \(\chi_{8041}(2323,\cdot)\) \(\chi_{8041}(2538,\cdot)\) \(\chi_{8041}(2581,\cdot)\) \(\chi_{8041}(2624,\cdot)\) \(\chi_{8041}(3054,\cdot)\) \(\chi_{8041}(3097,\cdot)\) \(\chi_{8041}(3269,\cdot)\) \(\chi_{8041}(4000,\cdot)\) \(\chi_{8041}(4043,\cdot)\) \(\chi_{8041}(4430,\cdot)\) \(\chi_{8041}(4516,\cdot)\) \(\chi_{8041}(4903,\cdot)\) \(\chi_{8041}(5161,\cdot)\) \(\chi_{8041}(5462,\cdot)\) \(\chi_{8041}(5634,\cdot)\) \(\chi_{8041}(5892,\cdot)\) \(\chi_{8041}(6365,\cdot)\) \(\chi_{8041}(6795,\cdot)\) \(\chi_{8041}(7354,\cdot)\) \(\chi_{8041}(7526,\cdot)\) ...

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: $\Q(\zeta_{80})$
Fixed field: Number field defined by a degree 80 polynomial

Values on generators

\((6580,2366,562)\) → \((e\left(\frac{1}{10}\right),e\left(\frac{7}{16}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 8041 }(2323, a) \) \(1\)\(1\)\(e\left(\frac{9}{40}\right)\)\(e\left(\frac{19}{80}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{47}{80}\right)\)\(e\left(\frac{37}{80}\right)\)\(e\left(\frac{41}{80}\right)\)\(e\left(\frac{27}{40}\right)\)\(e\left(\frac{19}{40}\right)\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{11}{16}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8041 }(2323,a) \;\) at \(\;a = \) e.g. 2