Properties

Label 6003.8
Modulus $6003$
Conductor $2001$
Order $308$
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([154,84,33]))
 
pari: [g,chi] = znchar(Mod(8,6003))
 

Basic properties

Modulus: \(6003\)
Conductor: \(2001\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(308\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2001}(8,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6003.dg

\(\chi_{6003}(8,\cdot)\) \(\chi_{6003}(26,\cdot)\) \(\chi_{6003}(98,\cdot)\) \(\chi_{6003}(188,\cdot)\) \(\chi_{6003}(242,\cdot)\) \(\chi_{6003}(269,\cdot)\) \(\chi_{6003}(305,\cdot)\) \(\chi_{6003}(395,\cdot)\) \(\chi_{6003}(404,\cdot)\) \(\chi_{6003}(449,\cdot)\) \(\chi_{6003}(485,\cdot)\) \(\chi_{6003}(512,\cdot)\) \(\chi_{6003}(611,\cdot)\) \(\chi_{6003}(656,\cdot)\) \(\chi_{6003}(791,\cdot)\) \(\chi_{6003}(809,\cdot)\) \(\chi_{6003}(926,\cdot)\) \(\chi_{6003}(1007,\cdot)\) \(\chi_{6003}(1025,\cdot)\) \(\chi_{6003}(1070,\cdot)\) \(\chi_{6003}(1133,\cdot)\) \(\chi_{6003}(1232,\cdot)\) \(\chi_{6003}(1250,\cdot)\) \(\chi_{6003}(1268,\cdot)\) \(\chi_{6003}(1313,\cdot)\) \(\chi_{6003}(1439,\cdot)\) \(\chi_{6003}(1511,\cdot)\) \(\chi_{6003}(1547,\cdot)\) \(\chi_{6003}(1664,\cdot)\) \(\chi_{6003}(1754,\cdot)\) ...

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

\((668,3133,4555)\) → \((-1,e\left(\frac{3}{11}\right),e\left(\frac{3}{28}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(1\)\(1\)\(e\left(\frac{47}{308}\right)\)\(e\left(\frac{47}{154}\right)\)\(e\left(\frac{10}{77}\right)\)\(e\left(\frac{36}{77}\right)\)\(e\left(\frac{141}{308}\right)\)\(e\left(\frac{87}{308}\right)\)\(e\left(\frac{195}{308}\right)\)\(e\left(\frac{115}{154}\right)\)\(e\left(\frac{191}{308}\right)\)\(e\left(\frac{47}{77}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{308})$
Fixed field: Number field defined by a degree 308 polynomial