Properties

Label 4008.1397
Modulus $4008$
Conductor $4008$
Order $166$
Real no
Primitive yes
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([0,83,83,66]))
 
pari: [g,chi] = znchar(Mod(1397,4008))
 

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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4008.r

\(\chi_{4008}(29,\cdot)\) \(\chi_{4008}(77,\cdot)\) \(\chi_{4008}(173,\cdot)\) \(\chi_{4008}(221,\cdot)\) \(\chi_{4008}(293,\cdot)\) \(\chi_{4008}(317,\cdot)\) \(\chi_{4008}(341,\cdot)\) \(\chi_{4008}(365,\cdot)\) \(\chi_{4008}(461,\cdot)\) \(\chi_{4008}(509,\cdot)\) \(\chi_{4008}(533,\cdot)\) \(\chi_{4008}(557,\cdot)\) \(\chi_{4008}(629,\cdot)\) \(\chi_{4008}(653,\cdot)\) \(\chi_{4008}(677,\cdot)\) \(\chi_{4008}(701,\cdot)\) \(\chi_{4008}(725,\cdot)\) \(\chi_{4008}(749,\cdot)\) \(\chi_{4008}(893,\cdot)\) \(\chi_{4008}(965,\cdot)\) \(\chi_{4008}(989,\cdot)\) \(\chi_{4008}(1013,\cdot)\) \(\chi_{4008}(1109,\cdot)\) \(\chi_{4008}(1181,\cdot)\) \(\chi_{4008}(1205,\cdot)\) \(\chi_{4008}(1253,\cdot)\) \(\chi_{4008}(1277,\cdot)\) \(\chi_{4008}(1301,\cdot)\) \(\chi_{4008}(1397,\cdot)\) \(\chi_{4008}(1421,\cdot)\) ...

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

\((3007,2005,1337,673)\) → \((1,-1,-1,e\left(\frac{33}{83}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 4008 }(1397, a) \) \(-1\)\(1\)\(e\left(\frac{33}{83}\right)\)\(e\left(\frac{76}{83}\right)\)\(e\left(\frac{11}{83}\right)\)\(e\left(\frac{75}{166}\right)\)\(e\left(\frac{95}{166}\right)\)\(e\left(\frac{93}{166}\right)\)\(e\left(\frac{143}{166}\right)\)\(e\left(\frac{66}{83}\right)\)\(e\left(\frac{53}{83}\right)\)\(e\left(\frac{65}{83}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4008 }(1397,a) \;\) at \(\;a = \) e.g. 2