Properties

Label 3549.131
Modulus $3549$
Conductor $3549$
Order $78$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(3549, base_ring=CyclotomicField(78))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([39,65,72]))
 
pari: [g,chi] = znchar(Mod(131,3549))
 

Basic properties

Modulus: \(3549\)
Conductor: \(3549\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(78\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3549.di

\(\chi_{3549}(131,\cdot)\) \(\chi_{3549}(248,\cdot)\) \(\chi_{3549}(404,\cdot)\) \(\chi_{3549}(521,\cdot)\) \(\chi_{3549}(794,\cdot)\) \(\chi_{3549}(950,\cdot)\) \(\chi_{3549}(1067,\cdot)\) \(\chi_{3549}(1223,\cdot)\) \(\chi_{3549}(1340,\cdot)\) \(\chi_{3549}(1496,\cdot)\) \(\chi_{3549}(1613,\cdot)\) \(\chi_{3549}(1769,\cdot)\) \(\chi_{3549}(1886,\cdot)\) \(\chi_{3549}(2042,\cdot)\) \(\chi_{3549}(2159,\cdot)\) \(\chi_{3549}(2315,\cdot)\) \(\chi_{3549}(2432,\cdot)\) \(\chi_{3549}(2588,\cdot)\) \(\chi_{3549}(2861,\cdot)\) \(\chi_{3549}(2978,\cdot)\) \(\chi_{3549}(3134,\cdot)\) \(\chi_{3549}(3251,\cdot)\) \(\chi_{3549}(3407,\cdot)\) \(\chi_{3549}(3524,\cdot)\)

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: $\Q(\zeta_{39})$
Fixed field: Number field defined by a degree 78 polynomial

Values on generators

\((1184,1522,3382)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{12}{13}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(16\)\(17\)\(19\)\(20\)
\(1\)\(1\)\(e\left(\frac{7}{78}\right)\)\(e\left(\frac{7}{39}\right)\)\(e\left(\frac{38}{39}\right)\)\(e\left(\frac{7}{26}\right)\)\(e\left(\frac{5}{78}\right)\)\(e\left(\frac{71}{78}\right)\)\(e\left(\frac{14}{39}\right)\)\(e\left(\frac{4}{39}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{2}{13}\right)\)
value at e.g. 2