Properties

Label 6003.349
Modulus $6003$
Conductor $207$
Order $33$
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([22,6,0]))
 
pari: [g,chi] = znchar(Mod(349,6003))
 

Basic properties

Modulus: \(6003\)
Conductor: \(207\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(33\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{207}(142,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6003.bt

\(\chi_{6003}(349,\cdot)\) \(\chi_{6003}(610,\cdot)\) \(\chi_{6003}(1393,\cdot)\) \(\chi_{6003}(1480,\cdot)\) \(\chi_{6003}(1741,\cdot)\) \(\chi_{6003}(1915,\cdot)\) \(\chi_{6003}(2263,\cdot)\) \(\chi_{6003}(2785,\cdot)\) \(\chi_{6003}(3307,\cdot)\) \(\chi_{6003}(3481,\cdot)\) \(\chi_{6003}(3568,\cdot)\) \(\chi_{6003}(3742,\cdot)\) \(\chi_{6003}(4264,\cdot)\) \(\chi_{6003}(4351,\cdot)\) \(\chi_{6003}(4612,\cdot)\) \(\chi_{6003}(4786,\cdot)\) \(\chi_{6003}(5308,\cdot)\) \(\chi_{6003}(5395,\cdot)\) \(\chi_{6003}(5569,\cdot)\) \(\chi_{6003}(5917,\cdot)\)

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

\((668,3133,4555)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{2}{11}\right),1)\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(1\)\(1\)\(e\left(\frac{1}{33}\right)\)\(e\left(\frac{2}{33}\right)\)\(e\left(\frac{17}{33}\right)\)\(e\left(\frac{4}{33}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{10}{33}\right)\)\(e\left(\frac{29}{33}\right)\)\(e\left(\frac{5}{33}\right)\)\(e\left(\frac{4}{33}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{33})\)
Fixed field: 33.33.70011645999218458416472683122408534303895571350166174758601569.1