Properties

Label 4008.511
Modulus $4008$
Conductor $668$
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(4008, base_ring=CyclotomicField(166))
 
M = H._module
 
chi = DirichletCharacter(H, M([83,0,0,41]))
 
pari: [g,chi] = znchar(Mod(511,4008))
 

Basic properties

Modulus: \(4008\)
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 \(\chi_{668}(511,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4008.bf

\(\chi_{4008}(55,\cdot)\) \(\chi_{4008}(79,\cdot)\) \(\chi_{4008}(103,\cdot)\) \(\chi_{4008}(151,\cdot)\) \(\chi_{4008}(247,\cdot)\) \(\chi_{4008}(271,\cdot)\) \(\chi_{4008}(439,\cdot)\) \(\chi_{4008}(463,\cdot)\) \(\chi_{4008}(487,\cdot)\) \(\chi_{4008}(511,\cdot)\) \(\chi_{4008}(535,\cdot)\) \(\chi_{4008}(583,\cdot)\) \(\chi_{4008}(607,\cdot)\) \(\chi_{4008}(703,\cdot)\) \(\chi_{4008}(727,\cdot)\) \(\chi_{4008}(751,\cdot)\) \(\chi_{4008}(799,\cdot)\) \(\chi_{4008}(823,\cdot)\) \(\chi_{4008}(895,\cdot)\) \(\chi_{4008}(991,\cdot)\) \(\chi_{4008}(1015,\cdot)\) \(\chi_{4008}(1039,\cdot)\) \(\chi_{4008}(1111,\cdot)\) \(\chi_{4008}(1255,\cdot)\) \(\chi_{4008}(1279,\cdot)\) \(\chi_{4008}(1303,\cdot)\) \(\chi_{4008}(1327,\cdot)\) \(\chi_{4008}(1351,\cdot)\) \(\chi_{4008}(1375,\cdot)\) \(\chi_{4008}(1447,\cdot)\) ...

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

\((3007,2005,1337,673)\) → \((-1,1,1,e\left(\frac{41}{166}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 4008 }(511, a) \) \(1\)\(1\)\(e\left(\frac{41}{166}\right)\)\(e\left(\frac{107}{166}\right)\)\(e\left(\frac{69}{166}\right)\)\(e\left(\frac{73}{166}\right)\)\(e\left(\frac{15}{166}\right)\)\(e\left(\frac{137}{166}\right)\)\(e\left(\frac{79}{83}\right)\)\(e\left(\frac{41}{83}\right)\)\(e\left(\frac{4}{83}\right)\)\(e\left(\frac{121}{166}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4008 }(511,a) \;\) at \(\;a = \) e.g. 2