Properties

Label 8041.100
Modulus $8041$
Conductor $731$
Order $168$
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(168))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,63,80]))
 
pari: [g,chi] = znchar(Mod(100,8041))
 

Basic properties

Modulus: \(8041\)
Conductor: \(731\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(168\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{731}(100,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8041.es

\(\chi_{8041}(100,\cdot)\) \(\chi_{8041}(111,\cdot)\) \(\chi_{8041}(144,\cdot)\) \(\chi_{8041}(298,\cdot)\) \(\chi_{8041}(529,\cdot)\) \(\chi_{8041}(705,\cdot)\) \(\chi_{8041}(848,\cdot)\) \(\chi_{8041}(1046,\cdot)\) \(\chi_{8041}(1090,\cdot)\) \(\chi_{8041}(1651,\cdot)\) \(\chi_{8041}(1794,\cdot)\) \(\chi_{8041}(2388,\cdot)\) \(\chi_{8041}(2575,\cdot)\) \(\chi_{8041}(2718,\cdot)\) \(\chi_{8041}(2762,\cdot)\) \(\chi_{8041}(2905,\cdot)\) \(\chi_{8041}(2949,\cdot)\) \(\chi_{8041}(3136,\cdot)\) \(\chi_{8041}(3334,\cdot)\) \(\chi_{8041}(3521,\cdot)\) \(\chi_{8041}(3664,\cdot)\) \(\chi_{8041}(3708,\cdot)\) \(\chi_{8041}(3840,\cdot)\) \(\chi_{8041}(3851,\cdot)\) \(\chi_{8041}(3884,\cdot)\) \(\chi_{8041}(3895,\cdot)\) \(\chi_{8041}(4082,\cdot)\) \(\chi_{8041}(4401,\cdot)\) \(\chi_{8041}(4632,\cdot)\) \(\chi_{8041}(4786,\cdot)\) ...

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: $\Q(\zeta_{168})$
Fixed field: Number field defined by a degree 168 polynomial (not computed)

Values on generators

\((6580,2366,562)\) → \((1,e\left(\frac{3}{8}\right),e\left(\frac{10}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 8041 }(100, a) \) \(1\)\(1\)\(e\left(\frac{3}{28}\right)\)\(e\left(\frac{143}{168}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{131}{168}\right)\)\(e\left(\frac{23}{24}\right)\)\(e\left(\frac{19}{24}\right)\)\(e\left(\frac{9}{28}\right)\)\(e\left(\frac{59}{84}\right)\)\(e\left(\frac{149}{168}\right)\)\(e\left(\frac{11}{168}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8041 }(100,a) \;\) at \(\;a = \) e.g. 2