Properties

Label 8041.1257
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([168,105,50]))
 
pari: [g,chi] = znchar(Mod(1257,8041))
 

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.fa

\(\chi_{8041}(152,\cdot)\) \(\chi_{8041}(169,\cdot)\) \(\chi_{8041}(203,\cdot)\) \(\chi_{8041}(339,\cdot)\) \(\chi_{8041}(526,\cdot)\) \(\chi_{8041}(713,\cdot)\) \(\chi_{8041}(883,\cdot)\) \(\chi_{8041}(900,\cdot)\) \(\chi_{8041}(917,\cdot)\) \(\chi_{8041}(1070,\cdot)\) \(\chi_{8041}(1257,\cdot)\) \(\chi_{8041}(1444,\cdot)\) \(\chi_{8041}(1631,\cdot)\) \(\chi_{8041}(1648,\cdot)\) \(\chi_{8041}(1665,\cdot)\) \(\chi_{8041}(2073,\cdot)\) \(\chi_{8041}(2260,\cdot)\) \(\chi_{8041}(2379,\cdot)\) \(\chi_{8041}(2396,\cdot)\) \(\chi_{8041}(3127,\cdot)\) \(\chi_{8041}(3195,\cdot)\) \(\chi_{8041}(3535,\cdot)\) \(\chi_{8041}(3722,\cdot)\) \(\chi_{8041}(3756,\cdot)\) \(\chi_{8041}(4266,\cdot)\) \(\chi_{8041}(4317,\cdot)\) \(\chi_{8041}(4453,\cdot)\) \(\chi_{8041}(4657,\cdot)\) \(\chi_{8041}(4997,\cdot)\) \(\chi_{8041}(5184,\cdot)\) ...

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

\((6580,2366,562)\) → \((e\left(\frac{4}{5}\right),-1,e\left(\frac{5}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 8041 }(1257, a) \) \(1\)\(1\)\(e\left(\frac{8}{35}\right)\)\(e\left(\frac{29}{210}\right)\)\(e\left(\frac{16}{35}\right)\)\(e\left(\frac{137}{210}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{24}{35}\right)\)\(e\left(\frac{29}{105}\right)\)\(e\left(\frac{37}{42}\right)\)\(e\left(\frac{25}{42}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8041 }(1257,a) \;\) at \(\;a = \) e.g. 2