Properties

Label 4235.1437
Modulus $4235$
Conductor $4235$
Order $660$
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(660))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([165,220,642]))
 
pari: [g,chi] = znchar(Mod(1437,4235))
 

Basic properties

Modulus: \(4235\)
Conductor: \(4235\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(660\)
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.do

\(\chi_{4235}(2,\cdot)\) \(\chi_{4235}(18,\cdot)\) \(\chi_{4235}(72,\cdot)\) \(\chi_{4235}(107,\cdot)\) \(\chi_{4235}(123,\cdot)\) \(\chi_{4235}(128,\cdot)\) \(\chi_{4235}(172,\cdot)\) \(\chi_{4235}(193,\cdot)\) \(\chi_{4235}(228,\cdot)\) \(\chi_{4235}(277,\cdot)\) \(\chi_{4235}(303,\cdot)\) \(\chi_{4235}(338,\cdot)\) \(\chi_{4235}(347,\cdot)\) \(\chi_{4235}(382,\cdot)\) \(\chi_{4235}(387,\cdot)\) \(\chi_{4235}(492,\cdot)\) \(\chi_{4235}(508,\cdot)\) \(\chi_{4235}(513,\cdot)\) \(\chi_{4235}(557,\cdot)\) \(\chi_{4235}(613,\cdot)\) \(\chi_{4235}(618,\cdot)\) \(\chi_{4235}(662,\cdot)\) \(\chi_{4235}(667,\cdot)\) \(\chi_{4235}(688,\cdot)\) \(\chi_{4235}(732,\cdot)\) \(\chi_{4235}(767,\cdot)\) \(\chi_{4235}(772,\cdot)\) \(\chi_{4235}(788,\cdot)\) \(\chi_{4235}(842,\cdot)\) \(\chi_{4235}(877,\cdot)\) ...

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: $\Q(\zeta_{660})$
Fixed field: Number field defined by a degree 660 polynomial (not computed)

Values on generators

\((2542,1816,2906)\) → \((i,e\left(\frac{1}{3}\right),e\left(\frac{107}{110}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(12\)\(13\)\(16\)\(17\)
\(1\)\(1\)\(e\left(\frac{587}{660}\right)\)\(e\left(\frac{41}{60}\right)\)\(e\left(\frac{257}{330}\right)\)\(e\left(\frac{63}{110}\right)\)\(e\left(\frac{147}{220}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{61}{132}\right)\)\(e\left(\frac{219}{220}\right)\)\(e\left(\frac{92}{165}\right)\)\(e\left(\frac{163}{660}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4235 }(1437,a) \;\) at \(\;a = \) e.g. 2