Properties

Label 3549.3242
Modulus $3549$
Conductor $507$
Order $52$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3549, base_ring=CyclotomicField(52))
 
M = H._module
 
chi = DirichletCharacter(H, M([26,0,7]))
 
pari: [g,chi] = znchar(Mod(3242,3549))
 

Basic properties

Modulus: \(3549\)
Conductor: \(507\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(52\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{507}(200,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3549.cw

\(\chi_{3549}(8,\cdot)\) \(\chi_{3549}(281,\cdot)\) \(\chi_{3549}(512,\cdot)\) \(\chi_{3549}(554,\cdot)\) \(\chi_{3549}(785,\cdot)\) \(\chi_{3549}(827,\cdot)\) \(\chi_{3549}(1058,\cdot)\) \(\chi_{3549}(1100,\cdot)\) \(\chi_{3549}(1331,\cdot)\) \(\chi_{3549}(1373,\cdot)\) \(\chi_{3549}(1604,\cdot)\) \(\chi_{3549}(1646,\cdot)\) \(\chi_{3549}(1877,\cdot)\) \(\chi_{3549}(1919,\cdot)\) \(\chi_{3549}(2150,\cdot)\) \(\chi_{3549}(2192,\cdot)\) \(\chi_{3549}(2423,\cdot)\) \(\chi_{3549}(2696,\cdot)\) \(\chi_{3549}(2738,\cdot)\) \(\chi_{3549}(2969,\cdot)\) \(\chi_{3549}(3011,\cdot)\) \(\chi_{3549}(3242,\cdot)\) \(\chi_{3549}(3284,\cdot)\) \(\chi_{3549}(3515,\cdot)\)

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: $\Q(\zeta_{52})$
Fixed field: Number field defined by a degree 52 polynomial

Values on generators

\((1184,1522,3382)\) → \((-1,1,e\left(\frac{7}{52}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 3549 }(3242, a) \) \(1\)\(1\)\(e\left(\frac{33}{52}\right)\)\(e\left(\frac{7}{26}\right)\)\(e\left(\frac{37}{52}\right)\)\(e\left(\frac{47}{52}\right)\)\(e\left(\frac{9}{26}\right)\)\(e\left(\frac{19}{52}\right)\)\(e\left(\frac{7}{13}\right)\)\(e\left(\frac{2}{13}\right)\)\(-i\)\(e\left(\frac{51}{52}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3549 }(3242,a) \;\) at \(\;a = \) e.g. 2