Properties

Label 3549.647
Modulus $3549$
Conductor $3549$
Order $78$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3549, base_ring=CyclotomicField(78))
 
M = H._module
 
chi = DirichletCharacter(H, M([39,13,59]))
 
pari: [g,chi] = znchar(Mod(647,3549))
 

Basic properties

Modulus: \(3549\)
Conductor: \(3549\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(78\)
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.dr

\(\chi_{3549}(101,\cdot)\) \(\chi_{3549}(173,\cdot)\) \(\chi_{3549}(374,\cdot)\) \(\chi_{3549}(446,\cdot)\) \(\chi_{3549}(647,\cdot)\) \(\chi_{3549}(719,\cdot)\) \(\chi_{3549}(920,\cdot)\) \(\chi_{3549}(1193,\cdot)\) \(\chi_{3549}(1265,\cdot)\) \(\chi_{3549}(1466,\cdot)\) \(\chi_{3549}(1538,\cdot)\) \(\chi_{3549}(1739,\cdot)\) \(\chi_{3549}(1811,\cdot)\) \(\chi_{3549}(2012,\cdot)\) \(\chi_{3549}(2084,\cdot)\) \(\chi_{3549}(2285,\cdot)\) \(\chi_{3549}(2357,\cdot)\) \(\chi_{3549}(2630,\cdot)\) \(\chi_{3549}(2831,\cdot)\) \(\chi_{3549}(2903,\cdot)\) \(\chi_{3549}(3104,\cdot)\) \(\chi_{3549}(3176,\cdot)\) \(\chi_{3549}(3377,\cdot)\) \(\chi_{3549}(3449,\cdot)\)

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: $\Q(\zeta_{39})$
Fixed field: Number field defined by a degree 78 polynomial

Values on generators

\((1184,1522,3382)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{59}{78}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 3549 }(647, a) \) \(1\)\(1\)\(e\left(\frac{23}{39}\right)\)\(e\left(\frac{7}{39}\right)\)\(e\left(\frac{11}{78}\right)\)\(e\left(\frac{10}{13}\right)\)\(e\left(\frac{19}{26}\right)\)\(e\left(\frac{1}{13}\right)\)\(e\left(\frac{14}{39}\right)\)\(e\left(\frac{4}{39}\right)\)\(1\)\(e\left(\frac{25}{78}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3549 }(647,a) \;\) at \(\;a = \) e.g. 2