Properties

Label 4008.857
Modulus $4008$
Conductor $501$
Order $166$
Real no
Primitive no
Minimal yes
Parity odd

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([0,0,83,68]))
 
pari: [g,chi] = znchar(Mod(857,4008))
 

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 \(\chi_{501}(356,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4008.ba

\(\chi_{4008}(65,\cdot)\) \(\chi_{4008}(89,\cdot)\) \(\chi_{4008}(137,\cdot)\) \(\chi_{4008}(185,\cdot)\) \(\chi_{4008}(209,\cdot)\) \(\chi_{4008}(233,\cdot)\) \(\chi_{4008}(281,\cdot)\) \(\chi_{4008}(329,\cdot)\) \(\chi_{4008}(353,\cdot)\) \(\chi_{4008}(449,\cdot)\) \(\chi_{4008}(545,\cdot)\) \(\chi_{4008}(617,\cdot)\) \(\chi_{4008}(689,\cdot)\) \(\chi_{4008}(761,\cdot)\) \(\chi_{4008}(809,\cdot)\) \(\chi_{4008}(857,\cdot)\) \(\chi_{4008}(929,\cdot)\) \(\chi_{4008}(1049,\cdot)\) \(\chi_{4008}(1193,\cdot)\) \(\chi_{4008}(1217,\cdot)\) \(\chi_{4008}(1241,\cdot)\) \(\chi_{4008}(1265,\cdot)\) \(\chi_{4008}(1313,\cdot)\) \(\chi_{4008}(1361,\cdot)\) \(\chi_{4008}(1385,\cdot)\) \(\chi_{4008}(1433,\cdot)\) \(\chi_{4008}(1457,\cdot)\) \(\chi_{4008}(1505,\cdot)\) \(\chi_{4008}(1553,\cdot)\) \(\chi_{4008}(1601,\cdot)\) ...

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

\((3007,2005,1337,673)\) → \((1,1,-1,e\left(\frac{34}{83}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(-1\)\(1\)\(e\left(\frac{151}{166}\right)\)\(e\left(\frac{28}{83}\right)\)\(e\left(\frac{161}{166}\right)\)\(e\left(\frac{16}{83}\right)\)\(e\left(\frac{35}{166}\right)\)\(e\left(\frac{63}{83}\right)\)\(e\left(\frac{9}{166}\right)\)\(e\left(\frac{68}{83}\right)\)\(e\left(\frac{157}{166}\right)\)\(e\left(\frac{72}{83}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{83})$
Fixed field: Number field defined by a degree 166 polynomial