Properties

Label 3549.115
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([0,26,139]))
 
pari: [g,chi] = znchar(Mod(115,3549))
 

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 \(\chi_{1183}(115,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3549.eb

\(\chi_{3549}(115,\cdot)\) \(\chi_{3549}(124,\cdot)\) \(\chi_{3549}(262,\cdot)\) \(\chi_{3549}(292,\cdot)\) \(\chi_{3549}(388,\cdot)\) \(\chi_{3549}(397,\cdot)\) \(\chi_{3549}(535,\cdot)\) \(\chi_{3549}(565,\cdot)\) \(\chi_{3549}(661,\cdot)\) \(\chi_{3549}(670,\cdot)\) \(\chi_{3549}(808,\cdot)\) \(\chi_{3549}(838,\cdot)\) \(\chi_{3549}(943,\cdot)\) \(\chi_{3549}(1081,\cdot)\) \(\chi_{3549}(1111,\cdot)\) \(\chi_{3549}(1207,\cdot)\) \(\chi_{3549}(1216,\cdot)\) \(\chi_{3549}(1354,\cdot)\) \(\chi_{3549}(1384,\cdot)\) \(\chi_{3549}(1480,\cdot)\) \(\chi_{3549}(1489,\cdot)\) \(\chi_{3549}(1627,\cdot)\) \(\chi_{3549}(1657,\cdot)\) \(\chi_{3549}(1753,\cdot)\) \(\chi_{3549}(1762,\cdot)\) \(\chi_{3549}(1900,\cdot)\) \(\chi_{3549}(1930,\cdot)\) \(\chi_{3549}(2026,\cdot)\) \(\chi_{3549}(2035,\cdot)\) \(\chi_{3549}(2173,\cdot)\) ...

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

\((1184,1522,3382)\) → \((1,e\left(\frac{1}{6}\right),e\left(\frac{139}{156}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(16\)\(17\)\(19\)\(20\)
\(1\)\(1\)\(e\left(\frac{35}{156}\right)\)\(e\left(\frac{35}{78}\right)\)\(e\left(\frac{133}{156}\right)\)\(e\left(\frac{35}{52}\right)\)\(e\left(\frac{1}{13}\right)\)\(e\left(\frac{23}{52}\right)\)\(e\left(\frac{35}{39}\right)\)\(e\left(\frac{10}{39}\right)\)\(-i\)\(e\left(\frac{47}{156}\right)\)
value at e.g. 2