Properties

Label 1323.593
Modulus $1323$
Conductor $147$
Order $42$
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(1323, base_ring=CyclotomicField(42))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([21,29]))
 
pari: [g,chi] = znchar(Mod(593,1323))
 

Basic properties

Modulus: \(1323\)
Conductor: \(147\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{147}(5,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1323.bs

\(\chi_{1323}(26,\cdot)\) \(\chi_{1323}(269,\cdot)\) \(\chi_{1323}(404,\cdot)\) \(\chi_{1323}(458,\cdot)\) \(\chi_{1323}(593,\cdot)\) \(\chi_{1323}(647,\cdot)\) \(\chi_{1323}(782,\cdot)\) \(\chi_{1323}(836,\cdot)\) \(\chi_{1323}(971,\cdot)\) \(\chi_{1323}(1025,\cdot)\) \(\chi_{1323}(1160,\cdot)\) \(\chi_{1323}(1214,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{21})\)
Fixed field: \(\Q(\zeta_{147})^+\)

Values on generators

\((785,1081)\) → \((-1,e\left(\frac{29}{42}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\(1\)\(1\)\(e\left(\frac{19}{42}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{41}{42}\right)\)\(e\left(\frac{5}{42}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{1}{6}\right)\)
value at e.g. 2