Properties

Label 3549.31
Modulus $3549$
Conductor $1183$
Order $156$
Real no
Primitive no
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(156))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,26,21]))
 
pari: [g,chi] = znchar(Mod(31,3549))
 

Basic properties

Modulus: \(3549\)
Conductor: \(1183\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(156\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1183}(31,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3549.ej

\(\chi_{3549}(31,\cdot)\) \(\chi_{3549}(73,\cdot)\) \(\chi_{3549}(187,\cdot)\) \(\chi_{3549}(229,\cdot)\) \(\chi_{3549}(304,\cdot)\) \(\chi_{3549}(346,\cdot)\) \(\chi_{3549}(460,\cdot)\) \(\chi_{3549}(502,\cdot)\) \(\chi_{3549}(619,\cdot)\) \(\chi_{3549}(733,\cdot)\) \(\chi_{3549}(850,\cdot)\) \(\chi_{3549}(892,\cdot)\) \(\chi_{3549}(1006,\cdot)\) \(\chi_{3549}(1048,\cdot)\) \(\chi_{3549}(1123,\cdot)\) \(\chi_{3549}(1165,\cdot)\) \(\chi_{3549}(1279,\cdot)\) \(\chi_{3549}(1321,\cdot)\) \(\chi_{3549}(1396,\cdot)\) \(\chi_{3549}(1438,\cdot)\) \(\chi_{3549}(1552,\cdot)\) \(\chi_{3549}(1594,\cdot)\) \(\chi_{3549}(1669,\cdot)\) \(\chi_{3549}(1711,\cdot)\) \(\chi_{3549}(1825,\cdot)\) \(\chi_{3549}(1867,\cdot)\) \(\chi_{3549}(1942,\cdot)\) \(\chi_{3549}(1984,\cdot)\) \(\chi_{3549}(2140,\cdot)\) \(\chi_{3549}(2215,\cdot)\) ...

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

\((1184,1522,3382)\) → \((1,e\left(\frac{1}{6}\right),e\left(\frac{7}{52}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(16\)\(17\)\(19\)\(20\)
\(1\)\(1\)\(e\left(\frac{73}{156}\right)\)\(e\left(\frac{73}{78}\right)\)\(e\left(\frac{7}{156}\right)\)\(e\left(\frac{21}{52}\right)\)\(e\left(\frac{20}{39}\right)\)\(e\left(\frac{83}{156}\right)\)\(e\left(\frac{34}{39}\right)\)\(e\left(\frac{32}{39}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{51}{52}\right)\)
value at e.g. 2