Properties

Label 3549.1367
Modulus $3549$
Conductor $3549$
Order $156$
Real no
Primitive yes
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(156))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([78,52,133]))
 
pari: [g,chi] = znchar(Mod(1367,3549))
 

Basic properties

Modulus: \(3549\)
Conductor: \(3549\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(156\)
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.em

\(\chi_{3549}(2,\cdot)\) \(\chi_{3549}(32,\cdot)\) \(\chi_{3549}(128,\cdot)\) \(\chi_{3549}(137,\cdot)\) \(\chi_{3549}(275,\cdot)\) \(\chi_{3549}(305,\cdot)\) \(\chi_{3549}(401,\cdot)\) \(\chi_{3549}(410,\cdot)\) \(\chi_{3549}(548,\cdot)\) \(\chi_{3549}(578,\cdot)\) \(\chi_{3549}(674,\cdot)\) \(\chi_{3549}(683,\cdot)\) \(\chi_{3549}(821,\cdot)\) \(\chi_{3549}(851,\cdot)\) \(\chi_{3549}(947,\cdot)\) \(\chi_{3549}(956,\cdot)\) \(\chi_{3549}(1124,\cdot)\) \(\chi_{3549}(1220,\cdot)\) \(\chi_{3549}(1229,\cdot)\) \(\chi_{3549}(1367,\cdot)\) \(\chi_{3549}(1397,\cdot)\) \(\chi_{3549}(1493,\cdot)\) \(\chi_{3549}(1640,\cdot)\) \(\chi_{3549}(1670,\cdot)\) \(\chi_{3549}(1766,\cdot)\) \(\chi_{3549}(1775,\cdot)\) \(\chi_{3549}(1913,\cdot)\) \(\chi_{3549}(1943,\cdot)\) \(\chi_{3549}(2039,\cdot)\) \(\chi_{3549}(2048,\cdot)\) ...

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: $\Q(\zeta_{156})$
Fixed field: Number field defined by a degree 156 polynomial (not computed)

Values on generators

\((1184,1522,3382)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{133}{156}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(16\)\(17\)\(19\)\(20\)
\(1\)\(1\)\(e\left(\frac{1}{52}\right)\)\(e\left(\frac{1}{26}\right)\)\(e\left(\frac{131}{156}\right)\)\(e\left(\frac{3}{52}\right)\)\(e\left(\frac{67}{78}\right)\)\(e\left(\frac{101}{156}\right)\)\(e\left(\frac{1}{13}\right)\)\(e\left(\frac{4}{13}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{137}{156}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3549 }(1367,a) \;\) at \(\;a = \) e.g. 2