Properties

Label 4008.113
Modulus $4008$
Conductor $501$
Order $166$
Real no
Primitive no
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([0,0,83,73]))
 
pari: [g,chi] = znchar(Mod(113,4008))
 

Basic properties

Modulus: \(4008\)
Conductor: \(501\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(166\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{501}(113,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4008.t

\(\chi_{4008}(17,\cdot)\) \(\chi_{4008}(41,\cdot)\) \(\chi_{4008}(113,\cdot)\) \(\chi_{4008}(161,\cdot)\) \(\chi_{4008}(257,\cdot)\) \(\chi_{4008}(305,\cdot)\) \(\chi_{4008}(377,\cdot)\) \(\chi_{4008}(401,\cdot)\) \(\chi_{4008}(425,\cdot)\) \(\chi_{4008}(473,\cdot)\) \(\chi_{4008}(497,\cdot)\) \(\chi_{4008}(521,\cdot)\) \(\chi_{4008}(569,\cdot)\) \(\chi_{4008}(593,\cdot)\) \(\chi_{4008}(641,\cdot)\) \(\chi_{4008}(665,\cdot)\) \(\chi_{4008}(713,\cdot)\) \(\chi_{4008}(737,\cdot)\) \(\chi_{4008}(785,\cdot)\) \(\chi_{4008}(833,\cdot)\) \(\chi_{4008}(881,\cdot)\) \(\chi_{4008}(905,\cdot)\) \(\chi_{4008}(953,\cdot)\) \(\chi_{4008}(977,\cdot)\) \(\chi_{4008}(1025,\cdot)\) \(\chi_{4008}(1073,\cdot)\) \(\chi_{4008}(1097,\cdot)\) \(\chi_{4008}(1121,\cdot)\) \(\chi_{4008}(1145,\cdot)\) \(\chi_{4008}(1289,\cdot)\) ...

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

\((3007,2005,1337,673)\) → \((1,1,-1,e\left(\frac{73}{166}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 4008 }(113, a) \) \(1\)\(1\)\(e\left(\frac{78}{83}\right)\)\(e\left(\frac{74}{83}\right)\)\(e\left(\frac{135}{166}\right)\)\(e\left(\frac{49}{166}\right)\)\(e\left(\frac{67}{83}\right)\)\(e\left(\frac{42}{83}\right)\)\(e\left(\frac{3}{83}\right)\)\(e\left(\frac{73}{83}\right)\)\(e\left(\frac{77}{166}\right)\)\(e\left(\frac{48}{83}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4008 }(113,a) \;\) at \(\;a = \) e.g. 2