Properties

Label 4008.907
Modulus $4008$
Conductor $1336$
Order $166$
Real no
Primitive no
Minimal yes
Parity odd

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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([83,83,0,142]))
 
pari: [g,chi] = znchar(Mod(907,4008))
 

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 \(\chi_{1336}(907,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4008.bd

\(\chi_{4008}(19,\cdot)\) \(\chi_{4008}(115,\cdot)\) \(\chi_{4008}(211,\cdot)\) \(\chi_{4008}(283,\cdot)\) \(\chi_{4008}(355,\cdot)\) \(\chi_{4008}(427,\cdot)\) \(\chi_{4008}(475,\cdot)\) \(\chi_{4008}(523,\cdot)\) \(\chi_{4008}(595,\cdot)\) \(\chi_{4008}(715,\cdot)\) \(\chi_{4008}(859,\cdot)\) \(\chi_{4008}(883,\cdot)\) \(\chi_{4008}(907,\cdot)\) \(\chi_{4008}(931,\cdot)\) \(\chi_{4008}(979,\cdot)\) \(\chi_{4008}(1027,\cdot)\) \(\chi_{4008}(1051,\cdot)\) \(\chi_{4008}(1099,\cdot)\) \(\chi_{4008}(1123,\cdot)\) \(\chi_{4008}(1171,\cdot)\) \(\chi_{4008}(1219,\cdot)\) \(\chi_{4008}(1267,\cdot)\) \(\chi_{4008}(1291,\cdot)\) \(\chi_{4008}(1339,\cdot)\) \(\chi_{4008}(1363,\cdot)\) \(\chi_{4008}(1411,\cdot)\) \(\chi_{4008}(1435,\cdot)\) \(\chi_{4008}(1483,\cdot)\) \(\chi_{4008}(1507,\cdot)\) \(\chi_{4008}(1531,\cdot)\) ...

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

\((3007,2005,1337,673)\) → \((-1,-1,1,e\left(\frac{71}{83}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 4008 }(907, a) \) \(-1\)\(1\)\(e\left(\frac{59}{166}\right)\)\(e\left(\frac{73}{166}\right)\)\(e\left(\frac{79}{83}\right)\)\(e\left(\frac{101}{166}\right)\)\(e\left(\frac{28}{83}\right)\)\(e\left(\frac{51}{83}\right)\)\(e\left(\frac{31}{166}\right)\)\(e\left(\frac{59}{83}\right)\)\(e\left(\frac{135}{166}\right)\)\(e\left(\frac{81}{166}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4008 }(907,a) \;\) at \(\;a = \) e.g. 2