Properties

Label 3549.68
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([39,65,74]))
 
pari: [g,chi] = znchar(Mod(68,3549))
 

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.dy

\(\chi_{3549}(68,\cdot)\) \(\chi_{3549}(269,\cdot)\) \(\chi_{3549}(341,\cdot)\) \(\chi_{3549}(542,\cdot)\) \(\chi_{3549}(614,\cdot)\) \(\chi_{3549}(815,\cdot)\) \(\chi_{3549}(887,\cdot)\) \(\chi_{3549}(1088,\cdot)\) \(\chi_{3549}(1361,\cdot)\) \(\chi_{3549}(1433,\cdot)\) \(\chi_{3549}(1634,\cdot)\) \(\chi_{3549}(1706,\cdot)\) \(\chi_{3549}(1907,\cdot)\) \(\chi_{3549}(1979,\cdot)\) \(\chi_{3549}(2180,\cdot)\) \(\chi_{3549}(2252,\cdot)\) \(\chi_{3549}(2453,\cdot)\) \(\chi_{3549}(2525,\cdot)\) \(\chi_{3549}(2798,\cdot)\) \(\chi_{3549}(2999,\cdot)\) \(\chi_{3549}(3071,\cdot)\) \(\chi_{3549}(3272,\cdot)\) \(\chi_{3549}(3344,\cdot)\) \(\chi_{3549}(3545,\cdot)\)

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

\((1184,1522,3382)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{37}{39}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(16\)\(17\)\(19\)\(20\)
\(1\)\(1\)\(e\left(\frac{3}{26}\right)\)\(e\left(\frac{3}{13}\right)\)\(e\left(\frac{8}{39}\right)\)\(e\left(\frac{9}{26}\right)\)\(e\left(\frac{25}{78}\right)\)\(e\left(\frac{43}{78}\right)\)\(e\left(\frac{6}{13}\right)\)\(e\left(\frac{11}{13}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{17}{39}\right)\)
value at e.g. 2