Properties

Label 3549.76
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([0,78,67]))
 
pari: [g,chi] = znchar(Mod(76,3549))
 

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 \(\chi_{1183}(76,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3549.ei

\(\chi_{3549}(76,\cdot)\) \(\chi_{3549}(97,\cdot)\) \(\chi_{3549}(202,\cdot)\) \(\chi_{3549}(223,\cdot)\) \(\chi_{3549}(349,\cdot)\) \(\chi_{3549}(370,\cdot)\) \(\chi_{3549}(475,\cdot)\) \(\chi_{3549}(496,\cdot)\) \(\chi_{3549}(622,\cdot)\) \(\chi_{3549}(643,\cdot)\) \(\chi_{3549}(748,\cdot)\) \(\chi_{3549}(769,\cdot)\) \(\chi_{3549}(895,\cdot)\) \(\chi_{3549}(916,\cdot)\) \(\chi_{3549}(1021,\cdot)\) \(\chi_{3549}(1042,\cdot)\) \(\chi_{3549}(1168,\cdot)\) \(\chi_{3549}(1189,\cdot)\) \(\chi_{3549}(1294,\cdot)\) \(\chi_{3549}(1315,\cdot)\) \(\chi_{3549}(1462,\cdot)\) \(\chi_{3549}(1567,\cdot)\) \(\chi_{3549}(1588,\cdot)\) \(\chi_{3549}(1714,\cdot)\) \(\chi_{3549}(1735,\cdot)\) \(\chi_{3549}(1861,\cdot)\) \(\chi_{3549}(1987,\cdot)\) \(\chi_{3549}(2008,\cdot)\) \(\chi_{3549}(2113,\cdot)\) \(\chi_{3549}(2134,\cdot)\) ...

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

\((1184,1522,3382)\) → \((1,-1,e\left(\frac{67}{156}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(16\)\(17\)\(19\)\(20\)
\(1\)\(1\)\(e\left(\frac{67}{156}\right)\)\(e\left(\frac{67}{78}\right)\)\(e\left(\frac{19}{52}\right)\)\(e\left(\frac{15}{52}\right)\)\(e\left(\frac{31}{39}\right)\)\(e\left(\frac{37}{156}\right)\)\(e\left(\frac{28}{39}\right)\)\(e\left(\frac{8}{39}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{35}{156}\right)\)
value at e.g. 2