Properties

Label 6003.64
Modulus $6003$
Conductor $667$
Order $154$
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([0,84,33]))
 
pari: [g,chi] = znchar(Mod(64,6003))
 

Basic properties

Modulus: \(6003\)
Conductor: \(667\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(154\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{667}(64,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6003.cy

\(\chi_{6003}(64,\cdot)\) \(\chi_{6003}(100,\cdot)\) \(\chi_{6003}(154,\cdot)\) \(\chi_{6003}(325,\cdot)\) \(\chi_{6003}(361,\cdot)\) \(\chi_{6003}(370,\cdot)\) \(\chi_{6003}(469,\cdot)\) \(\chi_{6003}(676,\cdot)\) \(\chi_{6003}(883,\cdot)\) \(\chi_{6003}(892,\cdot)\) \(\chi_{6003}(991,\cdot)\) \(\chi_{6003}(1108,\cdot)\) \(\chi_{6003}(1135,\cdot)\) \(\chi_{6003}(1153,\cdot)\) \(\chi_{6003}(1198,\cdot)\) \(\chi_{6003}(1369,\cdot)\) \(\chi_{6003}(1396,\cdot)\) \(\chi_{6003}(1405,\cdot)\) \(\chi_{6003}(1513,\cdot)\) \(\chi_{6003}(1720,\cdot)\) \(\chi_{6003}(1774,\cdot)\) \(\chi_{6003}(1918,\cdot)\) \(\chi_{6003}(1927,\cdot)\) \(\chi_{6003}(1936,\cdot)\) \(\chi_{6003}(1981,\cdot)\) \(\chi_{6003}(2152,\cdot)\) \(\chi_{6003}(2188,\cdot)\) \(\chi_{6003}(2197,\cdot)\) \(\chi_{6003}(2440,\cdot)\) \(\chi_{6003}(2557,\cdot)\) ...

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

\((668,3133,4555)\) → \((1,e\left(\frac{6}{11}\right),e\left(\frac{3}{14}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(1\)\(1\)\(e\left(\frac{47}{154}\right)\)\(e\left(\frac{47}{77}\right)\)\(e\left(\frac{20}{77}\right)\)\(e\left(\frac{72}{77}\right)\)\(e\left(\frac{141}{154}\right)\)\(e\left(\frac{87}{154}\right)\)\(e\left(\frac{41}{154}\right)\)\(e\left(\frac{38}{77}\right)\)\(e\left(\frac{37}{154}\right)\)\(e\left(\frac{17}{77}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{77})$
Fixed field: Number field defined by a degree 154 polynomial