Properties

Label 3549.1331
Modulus $3549$
Conductor $507$
Order $52$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(3549, base_ring=CyclotomicField(52))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([26,0,51]))
 
pari: [g,chi] = znchar(Mod(1331,3549))
 

Basic properties

Modulus: \(3549\)
Conductor: \(507\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(52\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{507}(317,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3549.cw

\(\chi_{3549}(8,\cdot)\) \(\chi_{3549}(281,\cdot)\) \(\chi_{3549}(512,\cdot)\) \(\chi_{3549}(554,\cdot)\) \(\chi_{3549}(785,\cdot)\) \(\chi_{3549}(827,\cdot)\) \(\chi_{3549}(1058,\cdot)\) \(\chi_{3549}(1100,\cdot)\) \(\chi_{3549}(1331,\cdot)\) \(\chi_{3549}(1373,\cdot)\) \(\chi_{3549}(1604,\cdot)\) \(\chi_{3549}(1646,\cdot)\) \(\chi_{3549}(1877,\cdot)\) \(\chi_{3549}(1919,\cdot)\) \(\chi_{3549}(2150,\cdot)\) \(\chi_{3549}(2192,\cdot)\) \(\chi_{3549}(2423,\cdot)\) \(\chi_{3549}(2696,\cdot)\) \(\chi_{3549}(2738,\cdot)\) \(\chi_{3549}(2969,\cdot)\) \(\chi_{3549}(3011,\cdot)\) \(\chi_{3549}(3242,\cdot)\) \(\chi_{3549}(3284,\cdot)\) \(\chi_{3549}(3515,\cdot)\)

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: $\Q(\zeta_{52})$
Fixed field: Number field defined by a degree 52 polynomial

Values on generators

\((1184,1522,3382)\) → \((-1,1,e\left(\frac{51}{52}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(16\)\(17\)\(19\)\(20\)
\(1\)\(1\)\(e\left(\frac{25}{52}\right)\)\(e\left(\frac{25}{26}\right)\)\(e\left(\frac{17}{52}\right)\)\(e\left(\frac{23}{52}\right)\)\(e\left(\frac{21}{26}\right)\)\(e\left(\frac{27}{52}\right)\)\(e\left(\frac{12}{13}\right)\)\(e\left(\frac{9}{13}\right)\)\(-i\)\(e\left(\frac{15}{52}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3549 }(1331,a) \;\) at \(\;a = \) e.g. 2