Properties

Label 3549.79
Modulus $3549$
Conductor $1183$
Order $39$
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([0,26,12]))
 
pari: [g,chi] = znchar(Mod(79,3549))
 

Basic properties

Modulus: \(3549\)
Conductor: \(1183\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(39\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1183}(79,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3549.ct

\(\chi_{3549}(79,\cdot)\) \(\chi_{3549}(235,\cdot)\) \(\chi_{3549}(352,\cdot)\) \(\chi_{3549}(625,\cdot)\) \(\chi_{3549}(781,\cdot)\) \(\chi_{3549}(898,\cdot)\) \(\chi_{3549}(1054,\cdot)\) \(\chi_{3549}(1171,\cdot)\) \(\chi_{3549}(1327,\cdot)\) \(\chi_{3549}(1444,\cdot)\) \(\chi_{3549}(1600,\cdot)\) \(\chi_{3549}(1717,\cdot)\) \(\chi_{3549}(1873,\cdot)\) \(\chi_{3549}(1990,\cdot)\) \(\chi_{3549}(2146,\cdot)\) \(\chi_{3549}(2263,\cdot)\) \(\chi_{3549}(2419,\cdot)\) \(\chi_{3549}(2692,\cdot)\) \(\chi_{3549}(2809,\cdot)\) \(\chi_{3549}(2965,\cdot)\) \(\chi_{3549}(3082,\cdot)\) \(\chi_{3549}(3238,\cdot)\) \(\chi_{3549}(3355,\cdot)\) \(\chi_{3549}(3511,\cdot)\)

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: 39.39.1501310100540182816122902385277086845494955654696971364374430555896401217548869308777432830007194746769.1

Values on generators

\((1184,1522,3382)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{2}{13}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(16\)\(17\)\(19\)\(20\)
\(1\)\(1\)\(e\left(\frac{32}{39}\right)\)\(e\left(\frac{25}{39}\right)\)\(e\left(\frac{2}{39}\right)\)\(e\left(\frac{6}{13}\right)\)\(e\left(\frac{34}{39}\right)\)\(e\left(\frac{7}{39}\right)\)\(e\left(\frac{11}{39}\right)\)\(e\left(\frac{31}{39}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{9}{13}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3549 }(79,a) \;\) at \(\;a = \) e.g. 2