Properties

Label 3549.44
Modulus $3549$
Conductor $3549$
Order $156$
Real no
Primitive yes
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([78,52,105]))
 
pari: [g,chi] = znchar(Mod(44,3549))
 

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.ef

\(\chi_{3549}(44,\cdot)\) \(\chi_{3549}(86,\cdot)\) \(\chi_{3549}(200,\cdot)\) \(\chi_{3549}(242,\cdot)\) \(\chi_{3549}(317,\cdot)\) \(\chi_{3549}(359,\cdot)\) \(\chi_{3549}(473,\cdot)\) \(\chi_{3549}(515,\cdot)\) \(\chi_{3549}(590,\cdot)\) \(\chi_{3549}(632,\cdot)\) \(\chi_{3549}(788,\cdot)\) \(\chi_{3549}(863,\cdot)\) \(\chi_{3549}(905,\cdot)\) \(\chi_{3549}(1019,\cdot)\) \(\chi_{3549}(1061,\cdot)\) \(\chi_{3549}(1136,\cdot)\) \(\chi_{3549}(1178,\cdot)\) \(\chi_{3549}(1292,\cdot)\) \(\chi_{3549}(1334,\cdot)\) \(\chi_{3549}(1409,\cdot)\) \(\chi_{3549}(1565,\cdot)\) \(\chi_{3549}(1607,\cdot)\) \(\chi_{3549}(1682,\cdot)\) \(\chi_{3549}(1724,\cdot)\) \(\chi_{3549}(1838,\cdot)\) \(\chi_{3549}(1880,\cdot)\) \(\chi_{3549}(1955,\cdot)\) \(\chi_{3549}(1997,\cdot)\) \(\chi_{3549}(2111,\cdot)\) \(\chi_{3549}(2153,\cdot)\) ...

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

\((1184,1522,3382)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{35}{52}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(16\)\(17\)\(19\)\(20\)
\(1\)\(1\)\(e\left(\frac{131}{156}\right)\)\(e\left(\frac{53}{78}\right)\)\(e\left(\frac{35}{156}\right)\)\(e\left(\frac{27}{52}\right)\)\(e\left(\frac{5}{78}\right)\)\(e\left(\frac{25}{156}\right)\)\(e\left(\frac{14}{39}\right)\)\(e\left(\frac{4}{39}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{47}{52}\right)\)
value at e.g. 2