Properties

Label 4235.648
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([99,88,30]))
 
pari: [g,chi] = znchar(Mod(648,4235))
 

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.db

\(\chi_{4235}(32,\cdot)\) \(\chi_{4235}(142,\cdot)\) \(\chi_{4235}(263,\cdot)\) \(\chi_{4235}(373,\cdot)\) \(\chi_{4235}(417,\cdot)\) \(\chi_{4235}(527,\cdot)\) \(\chi_{4235}(648,\cdot)\) \(\chi_{4235}(758,\cdot)\) \(\chi_{4235}(802,\cdot)\) \(\chi_{4235}(912,\cdot)\) \(\chi_{4235}(1033,\cdot)\) \(\chi_{4235}(1143,\cdot)\) \(\chi_{4235}(1187,\cdot)\) \(\chi_{4235}(1297,\cdot)\) \(\chi_{4235}(1418,\cdot)\) \(\chi_{4235}(1528,\cdot)\) \(\chi_{4235}(1682,\cdot)\) \(\chi_{4235}(1803,\cdot)\) \(\chi_{4235}(1913,\cdot)\) \(\chi_{4235}(1957,\cdot)\) \(\chi_{4235}(2067,\cdot)\) \(\chi_{4235}(2188,\cdot)\) \(\chi_{4235}(2342,\cdot)\) \(\chi_{4235}(2452,\cdot)\) \(\chi_{4235}(2573,\cdot)\) \(\chi_{4235}(2683,\cdot)\) \(\chi_{4235}(2727,\cdot)\) \(\chi_{4235}(2837,\cdot)\) \(\chi_{4235}(2958,\cdot)\) \(\chi_{4235}(3068,\cdot)\) ...

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

\((2542,1816,2906)\) → \((-i,e\left(\frac{2}{3}\right),e\left(\frac{5}{22}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(12\)\(13\)\(16\)\(17\)
\(1\)\(1\)\(e\left(\frac{41}{132}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{41}{66}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{41}{44}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{71}{132}\right)\)\(e\left(\frac{9}{44}\right)\)\(e\left(\frac{8}{33}\right)\)\(e\left(\frac{73}{132}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4235 }(648,a) \;\) at \(\;a = \) e.g. 2