Properties

Label 3549.3067
Modulus $3549$
Conductor $169$
Order $26$
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(26))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,3]))
 
pari: [g,chi] = znchar(Mod(3067,3549))
 

Basic properties

Modulus: \(3549\)
Conductor: \(169\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(26\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{169}(25,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3549.co

\(\chi_{3549}(64,\cdot)\) \(\chi_{3549}(610,\cdot)\) \(\chi_{3549}(883,\cdot)\) \(\chi_{3549}(1156,\cdot)\) \(\chi_{3549}(1429,\cdot)\) \(\chi_{3549}(1702,\cdot)\) \(\chi_{3549}(1975,\cdot)\) \(\chi_{3549}(2248,\cdot)\) \(\chi_{3549}(2521,\cdot)\) \(\chi_{3549}(2794,\cdot)\) \(\chi_{3549}(3067,\cdot)\) \(\chi_{3549}(3340,\cdot)\)

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: \(\Q(\zeta_{13})\)
Fixed field: 26.26.3830224792147131369362629348887201408953937846517364173.1

Values on generators

\((1184,1522,3382)\) → \((1,1,e\left(\frac{3}{26}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 3549 }(3067, a) \) \(1\)\(1\)\(e\left(\frac{3}{26}\right)\)\(e\left(\frac{3}{13}\right)\)\(e\left(\frac{1}{26}\right)\)\(e\left(\frac{9}{26}\right)\)\(e\left(\frac{2}{13}\right)\)\(e\left(\frac{23}{26}\right)\)\(e\left(\frac{6}{13}\right)\)\(e\left(\frac{11}{13}\right)\)\(-1\)\(e\left(\frac{7}{26}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3549 }(3067,a) \;\) at \(\;a = \) e.g. 2