Properties

Label 8016.2047
Modulus $8016$
Conductor $668$
Order $166$
Real no
Primitive no
Minimal yes
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([83,0,0,87]))
 
pari: [g,chi] = znchar(Mod(2047,8016))
 

Basic properties

Modulus: \(8016\)
Conductor: \(668\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(166\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{668}(43,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8016.bn

\(\chi_{8016}(79,\cdot)\) \(\chi_{8016}(271,\cdot)\) \(\chi_{8016}(463,\cdot)\) \(\chi_{8016}(511,\cdot)\) \(\chi_{8016}(607,\cdot)\) \(\chi_{8016}(703,\cdot)\) \(\chi_{8016}(751,\cdot)\) \(\chi_{8016}(799,\cdot)\) \(\chi_{8016}(895,\cdot)\) \(\chi_{8016}(991,\cdot)\) \(\chi_{8016}(1039,\cdot)\) \(\chi_{8016}(1279,\cdot)\) \(\chi_{8016}(1327,\cdot)\) \(\chi_{8016}(1375,\cdot)\) \(\chi_{8016}(1471,\cdot)\) \(\chi_{8016}(1663,\cdot)\) \(\chi_{8016}(1711,\cdot)\) \(\chi_{8016}(2047,\cdot)\) \(\chi_{8016}(2095,\cdot)\) \(\chi_{8016}(2143,\cdot)\) \(\chi_{8016}(2191,\cdot)\) \(\chi_{8016}(2239,\cdot)\) \(\chi_{8016}(2335,\cdot)\) \(\chi_{8016}(2383,\cdot)\) \(\chi_{8016}(2575,\cdot)\) \(\chi_{8016}(2623,\cdot)\) \(\chi_{8016}(2767,\cdot)\) \(\chi_{8016}(2815,\cdot)\) \(\chi_{8016}(2959,\cdot)\) \(\chi_{8016}(3151,\cdot)\) ...

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

\((3007,2005,5345,673)\) → \((-1,1,1,e\left(\frac{87}{166}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 8016 }(2047, a) \) \(1\)\(1\)\(e\left(\frac{87}{166}\right)\)\(e\left(\frac{57}{166}\right)\)\(e\left(\frac{29}{166}\right)\)\(e\left(\frac{163}{166}\right)\)\(e\left(\frac{129}{166}\right)\)\(e\left(\frac{149}{166}\right)\)\(e\left(\frac{32}{83}\right)\)\(e\left(\frac{4}{83}\right)\)\(e\left(\frac{51}{83}\right)\)\(e\left(\frac{111}{166}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8016 }(2047,a) \;\) at \(\;a = \) e.g. 2