Properties

Label 4008.685
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([0,83,0,53]))
 
pari: [g,chi] = znchar(Mod(685,4008))
 

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 \(\chi_{1336}(685,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4008.s

\(\chi_{4008}(13,\cdot)\) \(\chi_{4008}(37,\cdot)\) \(\chi_{4008}(109,\cdot)\) \(\chi_{4008}(253,\cdot)\) \(\chi_{4008}(277,\cdot)\) \(\chi_{4008}(301,\cdot)\) \(\chi_{4008}(325,\cdot)\) \(\chi_{4008}(349,\cdot)\) \(\chi_{4008}(373,\cdot)\) \(\chi_{4008}(445,\cdot)\) \(\chi_{4008}(469,\cdot)\) \(\chi_{4008}(493,\cdot)\) \(\chi_{4008}(541,\cdot)\) \(\chi_{4008}(637,\cdot)\) \(\chi_{4008}(661,\cdot)\) \(\chi_{4008}(685,\cdot)\) \(\chi_{4008}(709,\cdot)\) \(\chi_{4008}(781,\cdot)\) \(\chi_{4008}(829,\cdot)\) \(\chi_{4008}(925,\cdot)\) \(\chi_{4008}(973,\cdot)\) \(\chi_{4008}(1045,\cdot)\) \(\chi_{4008}(1069,\cdot)\) \(\chi_{4008}(1093,\cdot)\) \(\chi_{4008}(1141,\cdot)\) \(\chi_{4008}(1165,\cdot)\) \(\chi_{4008}(1189,\cdot)\) \(\chi_{4008}(1237,\cdot)\) \(\chi_{4008}(1261,\cdot)\) \(\chi_{4008}(1309,\cdot)\) ...

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

\((3007,2005,1337,673)\) → \((1,-1,1,e\left(\frac{53}{166}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 4008 }(685, a) \) \(-1\)\(1\)\(e\left(\frac{68}{83}\right)\)\(e\left(\frac{56}{83}\right)\)\(e\left(\frac{73}{166}\right)\)\(e\left(\frac{32}{83}\right)\)\(e\left(\frac{153}{166}\right)\)\(e\left(\frac{3}{166}\right)\)\(e\left(\frac{101}{166}\right)\)\(e\left(\frac{53}{83}\right)\)\(e\left(\frac{65}{166}\right)\)\(e\left(\frac{61}{83}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4008 }(685,a) \;\) at \(\;a = \) e.g. 2