Properties

Label 4008.673
Modulus $4008$
Conductor $167$
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([0,0,0,1]))
 
pari: [g,chi] = znchar(Mod(673,4008))
 

Basic properties

Modulus: \(4008\)
Conductor: \(167\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(166\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{167}(5,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4008.z

\(\chi_{4008}(73,\cdot)\) \(\chi_{4008}(145,\cdot)\) \(\chi_{4008}(193,\cdot)\) \(\chi_{4008}(241,\cdot)\) \(\chi_{4008}(313,\cdot)\) \(\chi_{4008}(385,\cdot)\) \(\chi_{4008}(457,\cdot)\) \(\chi_{4008}(553,\cdot)\) \(\chi_{4008}(649,\cdot)\) \(\chi_{4008}(673,\cdot)\) \(\chi_{4008}(721,\cdot)\) \(\chi_{4008}(769,\cdot)\) \(\chi_{4008}(793,\cdot)\) \(\chi_{4008}(817,\cdot)\) \(\chi_{4008}(865,\cdot)\) \(\chi_{4008}(913,\cdot)\) \(\chi_{4008}(937,\cdot)\) \(\chi_{4008}(1057,\cdot)\) \(\chi_{4008}(1081,\cdot)\) \(\chi_{4008}(1105,\cdot)\) \(\chi_{4008}(1153,\cdot)\) \(\chi_{4008}(1249,\cdot)\) \(\chi_{4008}(1273,\cdot)\) \(\chi_{4008}(1441,\cdot)\) \(\chi_{4008}(1465,\cdot)\) \(\chi_{4008}(1489,\cdot)\) \(\chi_{4008}(1513,\cdot)\) \(\chi_{4008}(1537,\cdot)\) \(\chi_{4008}(1585,\cdot)\) \(\chi_{4008}(1609,\cdot)\) ...

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

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

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(-1\)\(1\)\(e\left(\frac{1}{166}\right)\)\(e\left(\frac{59}{83}\right)\)\(e\left(\frac{14}{83}\right)\)\(e\left(\frac{103}{166}\right)\)\(e\left(\frac{53}{166}\right)\)\(e\left(\frac{29}{83}\right)\)\(e\left(\frac{99}{166}\right)\)\(e\left(\frac{1}{83}\right)\)\(e\left(\frac{75}{83}\right)\)\(e\left(\frac{45}{83}\right)\)
value at e.g. 2

Related number fields

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