Properties

Label 4008.191
Modulus $4008$
Conductor $2004$
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([83,0,83,48]))
 
pari: [g,chi] = znchar(Mod(191,4008))
 

Basic properties

Modulus: \(4008\)
Conductor: \(2004\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(166\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2004}(191,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4008.bc

\(\chi_{4008}(47,\cdot)\) \(\chi_{4008}(191,\cdot)\) \(\chi_{4008}(215,\cdot)\) \(\chi_{4008}(239,\cdot)\) \(\chi_{4008}(263,\cdot)\) \(\chi_{4008}(311,\cdot)\) \(\chi_{4008}(359,\cdot)\) \(\chi_{4008}(383,\cdot)\) \(\chi_{4008}(431,\cdot)\) \(\chi_{4008}(455,\cdot)\) \(\chi_{4008}(503,\cdot)\) \(\chi_{4008}(551,\cdot)\) \(\chi_{4008}(599,\cdot)\) \(\chi_{4008}(623,\cdot)\) \(\chi_{4008}(671,\cdot)\) \(\chi_{4008}(695,\cdot)\) \(\chi_{4008}(743,\cdot)\) \(\chi_{4008}(767,\cdot)\) \(\chi_{4008}(815,\cdot)\) \(\chi_{4008}(839,\cdot)\) \(\chi_{4008}(863,\cdot)\) \(\chi_{4008}(911,\cdot)\) \(\chi_{4008}(935,\cdot)\) \(\chi_{4008}(959,\cdot)\) \(\chi_{4008}(1031,\cdot)\) \(\chi_{4008}(1079,\cdot)\) \(\chi_{4008}(1175,\cdot)\) \(\chi_{4008}(1223,\cdot)\) \(\chi_{4008}(1295,\cdot)\) \(\chi_{4008}(1319,\cdot)\) ...

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

\((3007,2005,1337,673)\) → \((-1,1,-1,e\left(\frac{24}{83}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 4008 }(191, a) \) \(1\)\(1\)\(e\left(\frac{131}{166}\right)\)\(e\left(\frac{103}{166}\right)\)\(e\left(\frac{8}{83}\right)\)\(e\left(\frac{65}{83}\right)\)\(e\left(\frac{137}{166}\right)\)\(e\left(\frac{45}{166}\right)\)\(e\left(\frac{52}{83}\right)\)\(e\left(\frac{48}{83}\right)\)\(e\left(\frac{145}{166}\right)\)\(e\left(\frac{87}{166}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4008 }(191,a) \;\) at \(\;a = \) e.g. 2