Properties

Label 4008.605
Modulus $4008$
Conductor $4008$
Order $166$
Real no
Primitive yes
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([0,83,83,57]))
 
pari: [g,chi] = znchar(Mod(605,4008))
 

Basic properties

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

Galois orbit 4008.y

\(\chi_{4008}(5,\cdot)\) \(\chi_{4008}(53,\cdot)\) \(\chi_{4008}(101,\cdot)\) \(\chi_{4008}(125,\cdot)\) \(\chi_{4008}(149,\cdot)\) \(\chi_{4008}(197,\cdot)\) \(\chi_{4008}(245,\cdot)\) \(\chi_{4008}(269,\cdot)\) \(\chi_{4008}(389,\cdot)\) \(\chi_{4008}(413,\cdot)\) \(\chi_{4008}(437,\cdot)\) \(\chi_{4008}(485,\cdot)\) \(\chi_{4008}(581,\cdot)\) \(\chi_{4008}(605,\cdot)\) \(\chi_{4008}(773,\cdot)\) \(\chi_{4008}(797,\cdot)\) \(\chi_{4008}(821,\cdot)\) \(\chi_{4008}(845,\cdot)\) \(\chi_{4008}(869,\cdot)\) \(\chi_{4008}(917,\cdot)\) \(\chi_{4008}(941,\cdot)\) \(\chi_{4008}(1037,\cdot)\) \(\chi_{4008}(1061,\cdot)\) \(\chi_{4008}(1085,\cdot)\) \(\chi_{4008}(1133,\cdot)\) \(\chi_{4008}(1157,\cdot)\) \(\chi_{4008}(1229,\cdot)\) \(\chi_{4008}(1325,\cdot)\) \(\chi_{4008}(1349,\cdot)\) \(\chi_{4008}(1373,\cdot)\) ...

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

\((3007,2005,1337,673)\) → \((1,-1,-1,e\left(\frac{57}{166}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 4008 }(605, a) \) \(1\)\(1\)\(e\left(\frac{57}{166}\right)\)\(e\left(\frac{43}{83}\right)\)\(e\left(\frac{51}{83}\right)\)\(e\left(\frac{72}{83}\right)\)\(e\left(\frac{58}{83}\right)\)\(e\left(\frac{69}{166}\right)\)\(e\left(\frac{41}{83}\right)\)\(e\left(\frac{57}{83}\right)\)\(e\left(\frac{42}{83}\right)\)\(e\left(\frac{75}{83}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4008 }(605,a) \;\) at \(\;a = \) e.g. 2