Properties

Label 4235.628
Modulus $4235$
Conductor $4235$
Order $132$
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(4235, base_ring=CyclotomicField(132))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([99,110,84]))
 
pari: [g,chi] = znchar(Mod(628,4235))
 

Basic properties

Modulus: \(4235\)
Conductor: \(4235\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(132\)
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 4235.da

\(\chi_{4235}(12,\cdot)\) \(\chi_{4235}(353,\cdot)\) \(\chi_{4235}(397,\cdot)\) \(\chi_{4235}(507,\cdot)\) \(\chi_{4235}(628,\cdot)\) \(\chi_{4235}(738,\cdot)\) \(\chi_{4235}(782,\cdot)\) \(\chi_{4235}(892,\cdot)\) \(\chi_{4235}(1013,\cdot)\) \(\chi_{4235}(1123,\cdot)\) \(\chi_{4235}(1167,\cdot)\) \(\chi_{4235}(1277,\cdot)\) \(\chi_{4235}(1398,\cdot)\) \(\chi_{4235}(1508,\cdot)\) \(\chi_{4235}(1552,\cdot)\) \(\chi_{4235}(1662,\cdot)\) \(\chi_{4235}(1783,\cdot)\) \(\chi_{4235}(1893,\cdot)\) \(\chi_{4235}(2047,\cdot)\) \(\chi_{4235}(2168,\cdot)\) \(\chi_{4235}(2278,\cdot)\) \(\chi_{4235}(2322,\cdot)\) \(\chi_{4235}(2432,\cdot)\) \(\chi_{4235}(2553,\cdot)\) \(\chi_{4235}(2707,\cdot)\) \(\chi_{4235}(2817,\cdot)\) \(\chi_{4235}(2938,\cdot)\) \(\chi_{4235}(3048,\cdot)\) \(\chi_{4235}(3092,\cdot)\) \(\chi_{4235}(3202,\cdot)\) ...

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: $\Q(\zeta_{132})$
Fixed field: Number field defined by a degree 132 polynomial (not computed)

Values on generators

\((2542,1816,2906)\) → \((-i,e\left(\frac{5}{6}\right),e\left(\frac{7}{11}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(12\)\(13\)\(16\)\(17\)
\(1\)\(1\)\(e\left(\frac{7}{132}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{7}{66}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{7}{44}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{25}{132}\right)\)\(e\left(\frac{1}{44}\right)\)\(e\left(\frac{7}{33}\right)\)\(e\left(\frac{101}{132}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4235 }(628,a) \;\) at \(\;a = \) e.g. 2