Properties

Label 3549.101
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([39,13,35]))
 
pari: [g,chi] = znchar(Mod(101,3549))
 

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.dr

\(\chi_{3549}(101,\cdot)\) \(\chi_{3549}(173,\cdot)\) \(\chi_{3549}(374,\cdot)\) \(\chi_{3549}(446,\cdot)\) \(\chi_{3549}(647,\cdot)\) \(\chi_{3549}(719,\cdot)\) \(\chi_{3549}(920,\cdot)\) \(\chi_{3549}(1193,\cdot)\) \(\chi_{3549}(1265,\cdot)\) \(\chi_{3549}(1466,\cdot)\) \(\chi_{3549}(1538,\cdot)\) \(\chi_{3549}(1739,\cdot)\) \(\chi_{3549}(1811,\cdot)\) \(\chi_{3549}(2012,\cdot)\) \(\chi_{3549}(2084,\cdot)\) \(\chi_{3549}(2285,\cdot)\) \(\chi_{3549}(2357,\cdot)\) \(\chi_{3549}(2630,\cdot)\) \(\chi_{3549}(2831,\cdot)\) \(\chi_{3549}(2903,\cdot)\) \(\chi_{3549}(3104,\cdot)\) \(\chi_{3549}(3176,\cdot)\) \(\chi_{3549}(3377,\cdot)\) \(\chi_{3549}(3449,\cdot)\)

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

\((1184,1522,3382)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{35}{78}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(16\)\(17\)\(19\)\(20\)
\(1\)\(1\)\(e\left(\frac{11}{39}\right)\)\(e\left(\frac{22}{39}\right)\)\(e\left(\frac{29}{78}\right)\)\(e\left(\frac{11}{13}\right)\)\(e\left(\frac{17}{26}\right)\)\(e\left(\frac{5}{13}\right)\)\(e\left(\frac{5}{39}\right)\)\(e\left(\frac{7}{39}\right)\)\(1\)\(e\left(\frac{73}{78}\right)\)
value at e.g. 2