Properties

Label 4008.343
Modulus $4008$
Conductor $668$
Order $166$
Real no
Primitive no
Minimal no
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([83,0,0,22]))
 
pari: [g,chi] = znchar(Mod(343,4008))
 

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 \(\chi_{668}(343,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4008.w

\(\chi_{4008}(7,\cdot)\) \(\chi_{4008}(31,\cdot)\) \(\chi_{4008}(127,\cdot)\) \(\chi_{4008}(175,\cdot)\) \(\chi_{4008}(199,\cdot)\) \(\chi_{4008}(223,\cdot)\) \(\chi_{4008}(295,\cdot)\) \(\chi_{4008}(319,\cdot)\) \(\chi_{4008}(343,\cdot)\) \(\chi_{4008}(367,\cdot)\) \(\chi_{4008}(391,\cdot)\) \(\chi_{4008}(415,\cdot)\) \(\chi_{4008}(559,\cdot)\) \(\chi_{4008}(631,\cdot)\) \(\chi_{4008}(655,\cdot)\) \(\chi_{4008}(679,\cdot)\) \(\chi_{4008}(775,\cdot)\) \(\chi_{4008}(847,\cdot)\) \(\chi_{4008}(871,\cdot)\) \(\chi_{4008}(919,\cdot)\) \(\chi_{4008}(943,\cdot)\) \(\chi_{4008}(967,\cdot)\) \(\chi_{4008}(1063,\cdot)\) \(\chi_{4008}(1087,\cdot)\) \(\chi_{4008}(1135,\cdot)\) \(\chi_{4008}(1159,\cdot)\) \(\chi_{4008}(1183,\cdot)\) \(\chi_{4008}(1207,\cdot)\) \(\chi_{4008}(1231,\cdot)\) \(\chi_{4008}(1399,\cdot)\) ...

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

\((3007,2005,1337,673)\) → \((-1,1,1,e\left(\frac{11}{83}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 4008 }(343, a) \) \(-1\)\(1\)\(e\left(\frac{11}{83}\right)\)\(e\left(\frac{23}{166}\right)\)\(e\left(\frac{35}{166}\right)\)\(e\left(\frac{54}{83}\right)\)\(e\left(\frac{2}{83}\right)\)\(e\left(\frac{31}{166}\right)\)\(e\left(\frac{103}{166}\right)\)\(e\left(\frac{22}{83}\right)\)\(e\left(\frac{73}{83}\right)\)\(e\left(\frac{71}{166}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4008 }(343,a) \;\) at \(\;a = \) e.g. 2