Properties

Label 6003.320
Modulus $6003$
Conductor $207$
Order $66$
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(6003)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([55,39,0]))
 
pari: [g,chi] = znchar(Mod(320,6003))
 

Basic properties

Modulus: \(6003\)
Conductor: \(207\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{207}(113,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6003.ch

\(\chi_{6003}(320,\cdot)\) \(\chi_{6003}(842,\cdot)\) \(\chi_{6003}(1190,\cdot)\) \(\chi_{6003}(1364,\cdot)\) \(\chi_{6003}(1625,\cdot)\) \(\chi_{6003}(1712,\cdot)\) \(\chi_{6003}(2495,\cdot)\) \(\chi_{6003}(2756,\cdot)\) \(\chi_{6003}(3191,\cdot)\) \(\chi_{6003}(3539,\cdot)\) \(\chi_{6003}(3713,\cdot)\) \(\chi_{6003}(3800,\cdot)\) \(\chi_{6003}(4322,\cdot)\) \(\chi_{6003}(4496,\cdot)\) \(\chi_{6003}(4757,\cdot)\) \(\chi_{6003}(4844,\cdot)\) \(\chi_{6003}(5366,\cdot)\) \(\chi_{6003}(5540,\cdot)\) \(\chi_{6003}(5627,\cdot)\) \(\chi_{6003}(5801,\cdot)\)

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

\((668,3133,4555)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{13}{22}\right),1)\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(1\)\(1\)\(e\left(\frac{1}{66}\right)\)\(e\left(\frac{1}{33}\right)\)\(e\left(\frac{25}{33}\right)\)\(e\left(\frac{37}{66}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{5}{33}\right)\)\(e\left(\frac{31}{33}\right)\)\(e\left(\frac{19}{33}\right)\)\(e\left(\frac{2}{33}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 66 polynomial