Properties

Label 3549.2183
Modulus $3549$
Conductor $3549$
Order $26$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(3549, base_ring=CyclotomicField(26))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([13,13,5]))
 
pari: [g,chi] = znchar(Mod(2183,3549))
 

Basic properties

Modulus: \(3549\)
Conductor: \(3549\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(26\)
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.ck

\(\chi_{3549}(272,\cdot)\) \(\chi_{3549}(545,\cdot)\) \(\chi_{3549}(818,\cdot)\) \(\chi_{3549}(1091,\cdot)\) \(\chi_{3549}(1364,\cdot)\) \(\chi_{3549}(1637,\cdot)\) \(\chi_{3549}(1910,\cdot)\) \(\chi_{3549}(2183,\cdot)\) \(\chi_{3549}(2456,\cdot)\) \(\chi_{3549}(2729,\cdot)\) \(\chi_{3549}(3002,\cdot)\) \(\chi_{3549}(3275,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{13})\)
Fixed field: Number field defined by a degree 26 polynomial

Values on generators

\((1184,1522,3382)\) → \((-1,-1,e\left(\frac{5}{26}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(16\)\(17\)\(19\)\(20\)
\(1\)\(1\)\(e\left(\frac{9}{13}\right)\)\(e\left(\frac{5}{13}\right)\)\(e\left(\frac{19}{26}\right)\)\(e\left(\frac{1}{13}\right)\)\(e\left(\frac{11}{26}\right)\)\(e\left(\frac{4}{13}\right)\)\(e\left(\frac{10}{13}\right)\)\(e\left(\frac{1}{13}\right)\)\(1\)\(e\left(\frac{3}{26}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3549 }(2183,a) \;\) at \(\;a = \) e.g. 2