Properties

Label 4008.91
Modulus $4008$
Conductor $1336$
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(4008, base_ring=CyclotomicField(166))
 
M = H._module
 
chi = DirichletCharacter(H, M([83,83,0,55]))
 
pari: [g,chi] = znchar(Mod(91,4008))
 

Basic properties

Modulus: \(4008\)
Conductor: \(1336\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(166\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1336}(91,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4008.u

\(\chi_{4008}(43,\cdot)\) \(\chi_{4008}(67,\cdot)\) \(\chi_{4008}(91,\cdot)\) \(\chi_{4008}(139,\cdot)\) \(\chi_{4008}(163,\cdot)\) \(\chi_{4008}(187,\cdot)\) \(\chi_{4008}(235,\cdot)\) \(\chi_{4008}(259,\cdot)\) \(\chi_{4008}(307,\cdot)\) \(\chi_{4008}(331,\cdot)\) \(\chi_{4008}(379,\cdot)\) \(\chi_{4008}(403,\cdot)\) \(\chi_{4008}(451,\cdot)\) \(\chi_{4008}(499,\cdot)\) \(\chi_{4008}(547,\cdot)\) \(\chi_{4008}(571,\cdot)\) \(\chi_{4008}(619,\cdot)\) \(\chi_{4008}(643,\cdot)\) \(\chi_{4008}(691,\cdot)\) \(\chi_{4008}(739,\cdot)\) \(\chi_{4008}(763,\cdot)\) \(\chi_{4008}(787,\cdot)\) \(\chi_{4008}(811,\cdot)\) \(\chi_{4008}(955,\cdot)\) \(\chi_{4008}(1075,\cdot)\) \(\chi_{4008}(1147,\cdot)\) \(\chi_{4008}(1195,\cdot)\) \(\chi_{4008}(1243,\cdot)\) \(\chi_{4008}(1315,\cdot)\) \(\chi_{4008}(1387,\cdot)\) ...

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

\((3007,2005,1337,673)\) → \((-1,-1,1,e\left(\frac{55}{166}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 4008 }(91, a) \) \(1\)\(1\)\(e\left(\frac{69}{83}\right)\)\(e\left(\frac{99}{166}\right)\)\(e\left(\frac{23}{83}\right)\)\(e\left(\frac{52}{83}\right)\)\(e\left(\frac{93}{166}\right)\)\(e\left(\frac{18}{83}\right)\)\(e\left(\frac{25}{83}\right)\)\(e\left(\frac{55}{83}\right)\)\(e\left(\frac{33}{166}\right)\)\(e\left(\frac{53}{166}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4008 }(91,a) \;\) at \(\;a = \) e.g. 2