Properties

Label 3549.419
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([39,39,14]))
 
pari: [g,chi] = znchar(Mod(419,3549))
 

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.dc

\(\chi_{3549}(230,\cdot)\) \(\chi_{3549}(419,\cdot)\) \(\chi_{3549}(503,\cdot)\) \(\chi_{3549}(692,\cdot)\) \(\chi_{3549}(776,\cdot)\) \(\chi_{3549}(965,\cdot)\) \(\chi_{3549}(1049,\cdot)\) \(\chi_{3549}(1238,\cdot)\) \(\chi_{3549}(1322,\cdot)\) \(\chi_{3549}(1511,\cdot)\) \(\chi_{3549}(1595,\cdot)\) \(\chi_{3549}(1784,\cdot)\) \(\chi_{3549}(1868,\cdot)\) \(\chi_{3549}(2057,\cdot)\) \(\chi_{3549}(2141,\cdot)\) \(\chi_{3549}(2330,\cdot)\) \(\chi_{3549}(2414,\cdot)\) \(\chi_{3549}(2603,\cdot)\) \(\chi_{3549}(2687,\cdot)\) \(\chi_{3549}(2876,\cdot)\) \(\chi_{3549}(2960,\cdot)\) \(\chi_{3549}(3149,\cdot)\) \(\chi_{3549}(3422,\cdot)\) \(\chi_{3549}(3506,\cdot)\)

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

\((1184,1522,3382)\) → \((-1,-1,e\left(\frac{7}{39}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(16\)\(17\)\(19\)\(20\)
\(1\)\(1\)\(e\left(\frac{53}{78}\right)\)\(e\left(\frac{14}{39}\right)\)\(e\left(\frac{8}{13}\right)\)\(e\left(\frac{1}{26}\right)\)\(e\left(\frac{23}{78}\right)\)\(e\left(\frac{77}{78}\right)\)\(e\left(\frac{28}{39}\right)\)\(e\left(\frac{8}{39}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{38}{39}\right)\)
value at e.g. 2