Properties

Label 3549.2599
Modulus $3549$
Conductor $1183$
Order $78$
Real no
Primitive no
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(78))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,26,3]))
 
pari: [g,chi] = znchar(Mod(2599,3549))
 

Basic properties

Modulus: \(3549\)
Conductor: \(1183\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(78\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1183}(233,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3549.dg

\(\chi_{3549}(25,\cdot)\) \(\chi_{3549}(142,\cdot)\) \(\chi_{3549}(298,\cdot)\) \(\chi_{3549}(415,\cdot)\) \(\chi_{3549}(571,\cdot)\) \(\chi_{3549}(688,\cdot)\) \(\chi_{3549}(961,\cdot)\) \(\chi_{3549}(1117,\cdot)\) \(\chi_{3549}(1234,\cdot)\) \(\chi_{3549}(1390,\cdot)\) \(\chi_{3549}(1507,\cdot)\) \(\chi_{3549}(1663,\cdot)\) \(\chi_{3549}(1780,\cdot)\) \(\chi_{3549}(1936,\cdot)\) \(\chi_{3549}(2053,\cdot)\) \(\chi_{3549}(2209,\cdot)\) \(\chi_{3549}(2326,\cdot)\) \(\chi_{3549}(2482,\cdot)\) \(\chi_{3549}(2599,\cdot)\) \(\chi_{3549}(2755,\cdot)\) \(\chi_{3549}(3028,\cdot)\) \(\chi_{3549}(3145,\cdot)\) \(\chi_{3549}(3301,\cdot)\) \(\chi_{3549}(3418,\cdot)\)

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

Values on generators

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

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(16\)\(17\)\(19\)\(20\)
\(1\)\(1\)\(e\left(\frac{55}{78}\right)\)\(e\left(\frac{16}{39}\right)\)\(e\left(\frac{1}{78}\right)\)\(e\left(\frac{3}{26}\right)\)\(e\left(\frac{28}{39}\right)\)\(e\left(\frac{23}{78}\right)\)\(e\left(\frac{32}{39}\right)\)\(e\left(\frac{37}{39}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{11}{26}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3549 }(2599,a) \;\) at \(\;a = \) e.g. 2