Properties

Label 3549.2672
Modulus $3549$
Conductor $3549$
Order $156$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3549, base_ring=CyclotomicField(156))
 
M = H._module
 
chi = DirichletCharacter(H, M([78,130,83]))
 
pari: [g,chi] = znchar(Mod(2672,3549))
 

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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3549.ea

\(\chi_{3549}(110,\cdot)\) \(\chi_{3549}(206,\cdot)\) \(\chi_{3549}(215,\cdot)\) \(\chi_{3549}(353,\cdot)\) \(\chi_{3549}(383,\cdot)\) \(\chi_{3549}(479,\cdot)\) \(\chi_{3549}(626,\cdot)\) \(\chi_{3549}(656,\cdot)\) \(\chi_{3549}(752,\cdot)\) \(\chi_{3549}(761,\cdot)\) \(\chi_{3549}(899,\cdot)\) \(\chi_{3549}(929,\cdot)\) \(\chi_{3549}(1025,\cdot)\) \(\chi_{3549}(1034,\cdot)\) \(\chi_{3549}(1172,\cdot)\) \(\chi_{3549}(1298,\cdot)\) \(\chi_{3549}(1307,\cdot)\) \(\chi_{3549}(1445,\cdot)\) \(\chi_{3549}(1475,\cdot)\) \(\chi_{3549}(1571,\cdot)\) \(\chi_{3549}(1580,\cdot)\) \(\chi_{3549}(1718,\cdot)\) \(\chi_{3549}(1748,\cdot)\) \(\chi_{3549}(1844,\cdot)\) \(\chi_{3549}(1853,\cdot)\) \(\chi_{3549}(1991,\cdot)\) \(\chi_{3549}(2021,\cdot)\) \(\chi_{3549}(2126,\cdot)\) \(\chi_{3549}(2264,\cdot)\) \(\chi_{3549}(2294,\cdot)\) ...

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ 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

\((1184,1522,3382)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{83}{156}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 3549 }(2672, a) \) \(-1\)\(1\)\(e\left(\frac{109}{156}\right)\)\(e\left(\frac{31}{78}\right)\)\(e\left(\frac{71}{156}\right)\)\(e\left(\frac{5}{52}\right)\)\(e\left(\frac{2}{13}\right)\)\(e\left(\frac{33}{52}\right)\)\(e\left(\frac{31}{39}\right)\)\(e\left(\frac{1}{78}\right)\)\(-i\)\(e\left(\frac{133}{156}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3549 }(2672,a) \;\) at \(\;a = \) e.g. 2