Properties

Label 8043.76
Modulus $8043$
Conductor $2681$
Order $382$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8043, base_ring=CyclotomicField(382))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,191,16]))
 
pari: [g,chi] = znchar(Mod(76,8043))
 

Basic properties

Modulus: \(8043\)
Conductor: \(2681\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(382\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2681}(76,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8043.s

\(\chi_{8043}(34,\cdot)\) \(\chi_{8043}(55,\cdot)\) \(\chi_{8043}(76,\cdot)\) \(\chi_{8043}(139,\cdot)\) \(\chi_{8043}(202,\cdot)\) \(\chi_{8043}(223,\cdot)\) \(\chi_{8043}(265,\cdot)\) \(\chi_{8043}(286,\cdot)\) \(\chi_{8043}(370,\cdot)\) \(\chi_{8043}(391,\cdot)\) \(\chi_{8043}(412,\cdot)\) \(\chi_{8043}(433,\cdot)\) \(\chi_{8043}(454,\cdot)\) \(\chi_{8043}(475,\cdot)\) \(\chi_{8043}(496,\cdot)\) \(\chi_{8043}(517,\cdot)\) \(\chi_{8043}(643,\cdot)\) \(\chi_{8043}(706,\cdot)\) \(\chi_{8043}(727,\cdot)\) \(\chi_{8043}(769,\cdot)\) \(\chi_{8043}(790,\cdot)\) \(\chi_{8043}(853,\cdot)\) \(\chi_{8043}(874,\cdot)\) \(\chi_{8043}(895,\cdot)\) \(\chi_{8043}(916,\cdot)\) \(\chi_{8043}(937,\cdot)\) \(\chi_{8043}(958,\cdot)\) \(\chi_{8043}(979,\cdot)\) \(\chi_{8043}(1042,\cdot)\) \(\chi_{8043}(1105,\cdot)\) ...

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: $\Q(\zeta_{191})$
Fixed field: Number field defined by a degree 382 polynomial (not computed)

Values on generators

\((5363,2299,6133)\) → \((1,-1,e\left(\frac{8}{191}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 8043 }(76, a) \) \(-1\)\(1\)\(e\left(\frac{117}{191}\right)\)\(e\left(\frac{43}{191}\right)\)\(e\left(\frac{207}{382}\right)\)\(e\left(\frac{160}{191}\right)\)\(e\left(\frac{59}{382}\right)\)\(e\left(\frac{53}{191}\right)\)\(e\left(\frac{163}{382}\right)\)\(e\left(\frac{86}{191}\right)\)\(e\left(\frac{13}{382}\right)\)\(e\left(\frac{361}{382}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8043 }(76,a) \;\) at \(\;a = \) e.g. 2