Properties

Label 8041.1854
Modulus $8041$
Conductor $473$
Order $210$
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(210))
 
M = H._module
 
chi = DirichletCharacter(H, M([189,0,125]))
 
pari: [g,chi] = znchar(Mod(1854,8041))
 

Basic properties

Modulus: \(8041\)
Conductor: \(473\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(210\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{473}(435,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8041.fb

\(\chi_{8041}(18,\cdot)\) \(\chi_{8041}(205,\cdot)\) \(\chi_{8041}(392,\cdot)\) \(\chi_{8041}(545,\cdot)\) \(\chi_{8041}(579,\cdot)\) \(\chi_{8041}(800,\cdot)\) \(\chi_{8041}(1293,\cdot)\) \(\chi_{8041}(1361,\cdot)\) \(\chi_{8041}(1480,\cdot)\) \(\chi_{8041}(1531,\cdot)\) \(\chi_{8041}(1667,\cdot)\) \(\chi_{8041}(1854,\cdot)\) \(\chi_{8041}(1922,\cdot)\) \(\chi_{8041}(2041,\cdot)\) \(\chi_{8041}(2092,\cdot)\) \(\chi_{8041}(2262,\cdot)\) \(\chi_{8041}(2653,\cdot)\) \(\chi_{8041}(2823,\cdot)\) \(\chi_{8041}(2857,\cdot)\) \(\chi_{8041}(3044,\cdot)\) \(\chi_{8041}(3384,\cdot)\) \(\chi_{8041}(3588,\cdot)\) \(\chi_{8041}(3724,\cdot)\) \(\chi_{8041}(3775,\cdot)\) \(\chi_{8041}(4285,\cdot)\) \(\chi_{8041}(4319,\cdot)\) \(\chi_{8041}(4506,\cdot)\) \(\chi_{8041}(4846,\cdot)\) \(\chi_{8041}(4914,\cdot)\) \(\chi_{8041}(5645,\cdot)\) ...

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

\((6580,2366,562)\) → \((e\left(\frac{9}{10}\right),1,e\left(\frac{25}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 8041 }(1854, a) \) \(1\)\(1\)\(e\left(\frac{34}{35}\right)\)\(e\left(\frac{167}{210}\right)\)\(e\left(\frac{33}{35}\right)\)\(e\left(\frac{101}{210}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{32}{35}\right)\)\(e\left(\frac{62}{105}\right)\)\(e\left(\frac{19}{42}\right)\)\(e\left(\frac{31}{42}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8041 }(1854,a) \;\) at \(\;a = \) e.g. 2