Properties

Label 8016.1079
Modulus $8016$
Conductor $4008$
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([83,83,83,146]))
 
pari: [g,chi] = znchar(Mod(1079,8016))
 

Basic properties

Modulus: \(8016\)
Conductor: \(4008\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(166\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4008}(3083,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8016.bf

\(\chi_{8016}(215,\cdot)\) \(\chi_{8016}(263,\cdot)\) \(\chi_{8016}(311,\cdot)\) \(\chi_{8016}(359,\cdot)\) \(\chi_{8016}(455,\cdot)\) \(\chi_{8016}(503,\cdot)\) \(\chi_{8016}(551,\cdot)\) \(\chi_{8016}(599,\cdot)\) \(\chi_{8016}(695,\cdot)\) \(\chi_{8016}(743,\cdot)\) \(\chi_{8016}(839,\cdot)\) \(\chi_{8016}(935,\cdot)\) \(\chi_{8016}(1031,\cdot)\) \(\chi_{8016}(1079,\cdot)\) \(\chi_{8016}(1175,\cdot)\) \(\chi_{8016}(1223,\cdot)\) \(\chi_{8016}(1319,\cdot)\) \(\chi_{8016}(1367,\cdot)\) \(\chi_{8016}(1463,\cdot)\) \(\chi_{8016}(1511,\cdot)\) \(\chi_{8016}(1559,\cdot)\) \(\chi_{8016}(1655,\cdot)\) \(\chi_{8016}(1703,\cdot)\) \(\chi_{8016}(1751,\cdot)\) \(\chi_{8016}(1895,\cdot)\) \(\chi_{8016}(1991,\cdot)\) \(\chi_{8016}(2183,\cdot)\) \(\chi_{8016}(2279,\cdot)\) \(\chi_{8016}(2423,\cdot)\) \(\chi_{8016}(2471,\cdot)\) ...

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

\((3007,2005,5345,673)\) → \((-1,-1,-1,e\left(\frac{73}{83}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 8016 }(1079, a) \) \(1\)\(1\)\(e\left(\frac{73}{83}\right)\)\(e\left(\frac{47}{166}\right)\)\(e\left(\frac{21}{166}\right)\)\(e\left(\frac{15}{166}\right)\)\(e\left(\frac{19}{166}\right)\)\(e\left(\frac{1}{83}\right)\)\(e\left(\frac{6}{83}\right)\)\(e\left(\frac{63}{83}\right)\)\(e\left(\frac{77}{83}\right)\)\(e\left(\frac{109}{166}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8016 }(1079,a) \;\) at \(\;a = \) e.g. 2