Properties

Label 3549.50
Modulus $3549$
Conductor $507$
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([78,0,19]))
 
pari: [g,chi] = znchar(Mod(50,3549))
 

Basic properties

Modulus: \(3549\)
Conductor: \(507\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(156\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{507}(50,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3549.ee

\(\chi_{3549}(50,\cdot)\) \(\chi_{3549}(71,\cdot)\) \(\chi_{3549}(176,\cdot)\) \(\chi_{3549}(197,\cdot)\) \(\chi_{3549}(323,\cdot)\) \(\chi_{3549}(344,\cdot)\) \(\chi_{3549}(449,\cdot)\) \(\chi_{3549}(470,\cdot)\) \(\chi_{3549}(617,\cdot)\) \(\chi_{3549}(722,\cdot)\) \(\chi_{3549}(743,\cdot)\) \(\chi_{3549}(869,\cdot)\) \(\chi_{3549}(890,\cdot)\) \(\chi_{3549}(1016,\cdot)\) \(\chi_{3549}(1142,\cdot)\) \(\chi_{3549}(1163,\cdot)\) \(\chi_{3549}(1268,\cdot)\) \(\chi_{3549}(1289,\cdot)\) \(\chi_{3549}(1415,\cdot)\) \(\chi_{3549}(1436,\cdot)\) \(\chi_{3549}(1541,\cdot)\) \(\chi_{3549}(1562,\cdot)\) \(\chi_{3549}(1688,\cdot)\) \(\chi_{3549}(1814,\cdot)\) \(\chi_{3549}(1835,\cdot)\) \(\chi_{3549}(1961,\cdot)\) \(\chi_{3549}(1982,\cdot)\) \(\chi_{3549}(2087,\cdot)\) \(\chi_{3549}(2234,\cdot)\) \(\chi_{3549}(2255,\cdot)\) ...

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

\((1184,1522,3382)\) → \((-1,1,e\left(\frac{19}{156}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(16\)\(17\)\(19\)\(20\)
\(1\)\(1\)\(e\left(\frac{97}{156}\right)\)\(e\left(\frac{19}{78}\right)\)\(e\left(\frac{31}{52}\right)\)\(e\left(\frac{45}{52}\right)\)\(e\left(\frac{17}{78}\right)\)\(e\left(\frac{7}{156}\right)\)\(e\left(\frac{19}{39}\right)\)\(e\left(\frac{11}{39}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{131}{156}\right)\)
value at e.g. 2