Properties

Label 7098.5657
Modulus $7098$
Conductor $39$
Order $12$
Real no
Primitive no
Minimal no
Parity even

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(7098, base_ring=CyclotomicField(12))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([6,0,1]))
 
pari: [g,chi] = znchar(Mod(5657,7098))
 

Basic properties

Modulus: \(7098\)
Conductor: \(39\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{39}(2,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7098.bu

\(\chi_{7098}(995,\cdot)\) \(\chi_{7098}(1709,\cdot)\) \(\chi_{7098}(4145,\cdot)\) \(\chi_{7098}(5657,\cdot)\)

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: \(\Q(\zeta_{12})\)
Fixed field: \(\Q(\zeta_{39})^+\)

Values on generators

\((4733,5071,6931)\) → \((-1,1,e\left(\frac{1}{12}\right))\)

Values

\(-1\)\(1\)\(5\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\(1\)\(1\)\(i\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{1}{3}\right)\)\(-1\)\(e\left(\frac{5}{6}\right)\)\(-i\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{7}{12}\right)\)
value at e.g. 2