Properties

Label 3549.3461
Modulus $3549$
Conductor $3549$
Order $78$
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(78))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([39,13,14]))
 
pari: [g,chi] = znchar(Mod(3461,3549))
 

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.dk

\(\chi_{3549}(152,\cdot)\) \(\chi_{3549}(185,\cdot)\) \(\chi_{3549}(425,\cdot)\) \(\chi_{3549}(458,\cdot)\) \(\chi_{3549}(731,\cdot)\) \(\chi_{3549}(971,\cdot)\) \(\chi_{3549}(1004,\cdot)\) \(\chi_{3549}(1244,\cdot)\) \(\chi_{3549}(1277,\cdot)\) \(\chi_{3549}(1517,\cdot)\) \(\chi_{3549}(1550,\cdot)\) \(\chi_{3549}(1790,\cdot)\) \(\chi_{3549}(1823,\cdot)\) \(\chi_{3549}(2063,\cdot)\) \(\chi_{3549}(2096,\cdot)\) \(\chi_{3549}(2336,\cdot)\) \(\chi_{3549}(2369,\cdot)\) \(\chi_{3549}(2609,\cdot)\) \(\chi_{3549}(2642,\cdot)\) \(\chi_{3549}(2882,\cdot)\) \(\chi_{3549}(2915,\cdot)\) \(\chi_{3549}(3155,\cdot)\) \(\chi_{3549}(3428,\cdot)\) \(\chi_{3549}(3461,\cdot)\)

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

\((1184,1522,3382)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{7}{39}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(16\)\(17\)\(19\)\(20\)
\(1\)\(1\)\(e\left(\frac{1}{78}\right)\)\(e\left(\frac{1}{39}\right)\)\(e\left(\frac{37}{39}\right)\)\(e\left(\frac{1}{26}\right)\)\(e\left(\frac{25}{26}\right)\)\(e\left(\frac{17}{26}\right)\)\(e\left(\frac{2}{39}\right)\)\(e\left(\frac{34}{39}\right)\)\(-1\)\(e\left(\frac{38}{39}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3549 }(3461,a) \;\) at \(\;a = \) e.g. 2