Properties

Label 3549.2273
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([78,130,67]))
 
pari: [g,chi] = znchar(Mod(2273,3549))
 

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.ep

\(\chi_{3549}(59,\cdot)\) \(\chi_{3549}(227,\cdot)\) \(\chi_{3549}(236,\cdot)\) \(\chi_{3549}(332,\cdot)\) \(\chi_{3549}(362,\cdot)\) \(\chi_{3549}(500,\cdot)\) \(\chi_{3549}(509,\cdot)\) \(\chi_{3549}(605,\cdot)\) \(\chi_{3549}(635,\cdot)\) \(\chi_{3549}(773,\cdot)\) \(\chi_{3549}(782,\cdot)\) \(\chi_{3549}(878,\cdot)\) \(\chi_{3549}(908,\cdot)\) \(\chi_{3549}(1046,\cdot)\) \(\chi_{3549}(1055,\cdot)\) \(\chi_{3549}(1151,\cdot)\) \(\chi_{3549}(1181,\cdot)\) \(\chi_{3549}(1319,\cdot)\) \(\chi_{3549}(1328,\cdot)\) \(\chi_{3549}(1424,\cdot)\) \(\chi_{3549}(1454,\cdot)\) \(\chi_{3549}(1592,\cdot)\) \(\chi_{3549}(1697,\cdot)\) \(\chi_{3549}(1727,\cdot)\) \(\chi_{3549}(1865,\cdot)\) \(\chi_{3549}(1874,\cdot)\) \(\chi_{3549}(1970,\cdot)\) \(\chi_{3549}(2000,\cdot)\) \(\chi_{3549}(2138,\cdot)\) \(\chi_{3549}(2147,\cdot)\) ...

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

\((1184,1522,3382)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{67}{156}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 3549 }(2273, a) \) \(-1\)\(1\)\(e\left(\frac{31}{52}\right)\)\(e\left(\frac{5}{26}\right)\)\(e\left(\frac{83}{156}\right)\)\(e\left(\frac{41}{52}\right)\)\(e\left(\frac{5}{39}\right)\)\(e\left(\frac{11}{156}\right)\)\(e\left(\frac{5}{13}\right)\)\(e\left(\frac{1}{26}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{113}{156}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3549 }(2273,a) \;\) at \(\;a = \) e.g. 2