Properties

Label 8041.3416
Modulus $8041$
Conductor $8041$
Order $210$
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(210))
 
M = H._module
 
chi = DirichletCharacter(H, M([189,105,95]))
 
pari: [g,chi] = znchar(Mod(3416,8041))
 

Basic properties

Modulus: \(8041\)
Conductor: \(8041\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(210\)
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.ez

\(\chi_{8041}(288,\cdot)\) \(\chi_{8041}(458,\cdot)\) \(\chi_{8041}(492,\cdot)\) \(\chi_{8041}(679,\cdot)\) \(\chi_{8041}(1019,\cdot)\) \(\chi_{8041}(1223,\cdot)\) \(\chi_{8041}(1359,\cdot)\) \(\chi_{8041}(1410,\cdot)\) \(\chi_{8041}(1920,\cdot)\) \(\chi_{8041}(1954,\cdot)\) \(\chi_{8041}(2141,\cdot)\) \(\chi_{8041}(2481,\cdot)\) \(\chi_{8041}(2549,\cdot)\) \(\chi_{8041}(3280,\cdot)\) \(\chi_{8041}(3297,\cdot)\) \(\chi_{8041}(3416,\cdot)\) \(\chi_{8041}(3603,\cdot)\) \(\chi_{8041}(4011,\cdot)\) \(\chi_{8041}(4028,\cdot)\) \(\chi_{8041}(4045,\cdot)\) \(\chi_{8041}(4232,\cdot)\) \(\chi_{8041}(4419,\cdot)\) \(\chi_{8041}(4606,\cdot)\) \(\chi_{8041}(4759,\cdot)\) \(\chi_{8041}(4776,\cdot)\) \(\chi_{8041}(4793,\cdot)\) \(\chi_{8041}(4963,\cdot)\) \(\chi_{8041}(5150,\cdot)\) \(\chi_{8041}(5337,\cdot)\) \(\chi_{8041}(5473,\cdot)\) ...

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: $\Q(\zeta_{105})$
Fixed field: Number field defined by a degree 210 polynomial (not computed)

Values on generators

\((6580,2366,562)\) → \((e\left(\frac{9}{10}\right),-1,e\left(\frac{19}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 8041 }(3416, a) \) \(1\)\(1\)\(e\left(\frac{4}{35}\right)\)\(e\left(\frac{16}{105}\right)\)\(e\left(\frac{8}{35}\right)\)\(e\left(\frac{43}{105}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{12}{35}\right)\)\(e\left(\frac{32}{105}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{8}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8041 }(3416,a) \;\) at \(\;a = \) e.g. 2