Properties

Label 8016.1409
Modulus $8016$
Conductor $501$
Order $166$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8016, base_ring=CyclotomicField(166))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,83,89]))
 
pari: [g,chi] = znchar(Mod(1409,8016))
 

Basic properties

Modulus: \(8016\)
Conductor: \(501\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(166\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{501}(407,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8016.bb

\(\chi_{8016}(17,\cdot)\) \(\chi_{8016}(113,\cdot)\) \(\chi_{8016}(161,\cdot)\) \(\chi_{8016}(257,\cdot)\) \(\chi_{8016}(305,\cdot)\) \(\chi_{8016}(401,\cdot)\) \(\chi_{8016}(497,\cdot)\) \(\chi_{8016}(593,\cdot)\) \(\chi_{8016}(641,\cdot)\) \(\chi_{8016}(737,\cdot)\) \(\chi_{8016}(785,\cdot)\) \(\chi_{8016}(833,\cdot)\) \(\chi_{8016}(881,\cdot)\) \(\chi_{8016}(977,\cdot)\) \(\chi_{8016}(1025,\cdot)\) \(\chi_{8016}(1073,\cdot)\) \(\chi_{8016}(1121,\cdot)\) \(\chi_{8016}(1409,\cdot)\) \(\chi_{8016}(1649,\cdot)\) \(\chi_{8016}(1793,\cdot)\) \(\chi_{8016}(1889,\cdot)\) \(\chi_{8016}(1985,\cdot)\) \(\chi_{8016}(2129,\cdot)\) \(\chi_{8016}(2273,\cdot)\) \(\chi_{8016}(2417,\cdot)\) \(\chi_{8016}(2609,\cdot)\) \(\chi_{8016}(2801,\cdot)\) \(\chi_{8016}(2849,\cdot)\) \(\chi_{8016}(2945,\cdot)\) \(\chi_{8016}(3041,\cdot)\) ...

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: $\Q(\zeta_{83})$
Fixed field: Number field defined by a degree 166 polynomial (not computed)

Values on generators

\((3007,2005,5345,673)\) → \((1,1,-1,e\left(\frac{89}{166}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 8016 }(1409, a) \) \(1\)\(1\)\(e\left(\frac{3}{83}\right)\)\(e\left(\frac{22}{83}\right)\)\(e\left(\frac{85}{166}\right)\)\(e\left(\frac{37}{166}\right)\)\(e\left(\frac{76}{83}\right)\)\(e\left(\frac{8}{83}\right)\)\(e\left(\frac{48}{83}\right)\)\(e\left(\frac{6}{83}\right)\)\(e\left(\frac{153}{166}\right)\)\(e\left(\frac{21}{83}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8016 }(1409,a) \;\) at \(\;a = \) e.g. 2