Properties

Label 6003.80
Modulus $6003$
Conductor $2001$
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([77,63,143]))
 
pari: [g,chi] = znchar(Mod(80,6003))
 

Basic properties

Modulus: \(6003\)
Conductor: \(2001\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(154\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2001}(80,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6003.db

\(\chi_{6003}(80,\cdot)\) \(\chi_{6003}(125,\cdot)\) \(\chi_{6003}(296,\cdot)\) \(\chi_{6003}(332,\cdot)\) \(\chi_{6003}(341,\cdot)\) \(\chi_{6003}(557,\cdot)\) \(\chi_{6003}(701,\cdot)\) \(\chi_{6003}(845,\cdot)\) \(\chi_{6003}(908,\cdot)\) \(\chi_{6003}(962,\cdot)\) \(\chi_{6003}(1079,\cdot)\) \(\chi_{6003}(1115,\cdot)\) \(\chi_{6003}(1124,\cdot)\) \(\chi_{6003}(1169,\cdot)\) \(\chi_{6003}(1367,\cdot)\) \(\chi_{6003}(1376,\cdot)\) \(\chi_{6003}(1385,\cdot)\) \(\chi_{6003}(1601,\cdot)\) \(\chi_{6003}(1745,\cdot)\) \(\chi_{6003}(1907,\cdot)\) \(\chi_{6003}(1952,\cdot)\) \(\chi_{6003}(2006,\cdot)\) \(\chi_{6003}(2123,\cdot)\) \(\chi_{6003}(2150,\cdot)\) \(\chi_{6003}(2159,\cdot)\) \(\chi_{6003}(2213,\cdot)\) \(\chi_{6003}(2384,\cdot)\) \(\chi_{6003}(2411,\cdot)\) \(\chi_{6003}(2420,\cdot)\) \(\chi_{6003}(2429,\cdot)\) ...

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

\((668,3133,4555)\) → \((-1,e\left(\frac{9}{22}\right),e\left(\frac{13}{14}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(1\)\(1\)\(e\left(\frac{19}{77}\right)\)\(e\left(\frac{38}{77}\right)\)\(e\left(\frac{26}{77}\right)\)\(e\left(\frac{141}{154}\right)\)\(e\left(\frac{57}{77}\right)\)\(e\left(\frac{45}{77}\right)\)\(e\left(\frac{61}{154}\right)\)\(e\left(\frac{34}{77}\right)\)\(e\left(\frac{25}{154}\right)\)\(e\left(\frac{76}{77}\right)\)
value at e.g. 2

Related number fields

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