Properties

Label 4008.481
Modulus $4008$
Conductor $167$
Order $83$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(4008)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,0,0,82]))
 
pari: [g,chi] = znchar(Mod(481,4008))
 

Basic properties

Modulus: \(4008\)
Conductor: \(167\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(83\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{167}(147,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4008.q

\(\chi_{4008}(25,\cdot)\) \(\chi_{4008}(49,\cdot)\) \(\chi_{4008}(97,\cdot)\) \(\chi_{4008}(121,\cdot)\) \(\chi_{4008}(169,\cdot)\) \(\chi_{4008}(217,\cdot)\) \(\chi_{4008}(265,\cdot)\) \(\chi_{4008}(289,\cdot)\) \(\chi_{4008}(337,\cdot)\) \(\chi_{4008}(361,\cdot)\) \(\chi_{4008}(409,\cdot)\) \(\chi_{4008}(433,\cdot)\) \(\chi_{4008}(481,\cdot)\) \(\chi_{4008}(505,\cdot)\) \(\chi_{4008}(529,\cdot)\) \(\chi_{4008}(577,\cdot)\) \(\chi_{4008}(601,\cdot)\) \(\chi_{4008}(625,\cdot)\) \(\chi_{4008}(697,\cdot)\) \(\chi_{4008}(745,\cdot)\) \(\chi_{4008}(841,\cdot)\) \(\chi_{4008}(889,\cdot)\) \(\chi_{4008}(961,\cdot)\) \(\chi_{4008}(985,\cdot)\) \(\chi_{4008}(1009,\cdot)\) \(\chi_{4008}(1033,\cdot)\) \(\chi_{4008}(1129,\cdot)\) \(\chi_{4008}(1177,\cdot)\) \(\chi_{4008}(1201,\cdot)\) \(\chi_{4008}(1225,\cdot)\) ...

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Values on generators

\((3007,2005,1337,673)\) → \((1,1,1,e\left(\frac{82}{83}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(1\)\(1\)\(e\left(\frac{82}{83}\right)\)\(e\left(\frac{48}{83}\right)\)\(e\left(\frac{55}{83}\right)\)\(e\left(\frac{63}{83}\right)\)\(e\left(\frac{30}{83}\right)\)\(e\left(\frac{25}{83}\right)\)\(e\left(\frac{67}{83}\right)\)\(e\left(\frac{81}{83}\right)\)\(e\left(\frac{16}{83}\right)\)\(e\left(\frac{76}{83}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{83})$
Fixed field: Number field defined by a degree 83 polynomial