Properties

Label 4235.213
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([495,110,432]))
 
pari: [g,chi] = znchar(Mod(213,4235))
 

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.dp

\(\chi_{4235}(38,\cdot)\) \(\chi_{4235}(47,\cdot)\) \(\chi_{4235}(82,\cdot)\) \(\chi_{4235}(103,\cdot)\) \(\chi_{4235}(108,\cdot)\) \(\chi_{4235}(152,\cdot)\) \(\chi_{4235}(157,\cdot)\) \(\chi_{4235}(192,\cdot)\) \(\chi_{4235}(213,\cdot)\) \(\chi_{4235}(257,\cdot)\) \(\chi_{4235}(262,\cdot)\) \(\chi_{4235}(278,\cdot)\) \(\chi_{4235}(313,\cdot)\) \(\chi_{4235}(367,\cdot)\) \(\chi_{4235}(383,\cdot)\) \(\chi_{4235}(388,\cdot)\) \(\chi_{4235}(423,\cdot)\) \(\chi_{4235}(432,\cdot)\) \(\chi_{4235}(467,\cdot)\) \(\chi_{4235}(488,\cdot)\) \(\chi_{4235}(537,\cdot)\) \(\chi_{4235}(542,\cdot)\) \(\chi_{4235}(577,\cdot)\) \(\chi_{4235}(598,\cdot)\) \(\chi_{4235}(642,\cdot)\) \(\chi_{4235}(647,\cdot)\) \(\chi_{4235}(663,\cdot)\) \(\chi_{4235}(698,\cdot)\) \(\chi_{4235}(752,\cdot)\) \(\chi_{4235}(768,\cdot)\) ...

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

\((2542,1816,2906)\) → \((-i,e\left(\frac{1}{6}\right),e\left(\frac{36}{55}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(12\)\(13\)\(16\)\(17\)
\(1\)\(1\)\(e\left(\frac{487}{660}\right)\)\(e\left(\frac{1}{60}\right)\)\(e\left(\frac{157}{330}\right)\)\(e\left(\frac{83}{110}\right)\)\(e\left(\frac{47}{220}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{65}{132}\right)\)\(e\left(\frac{189}{220}\right)\)\(e\left(\frac{157}{165}\right)\)\(e\left(\frac{653}{660}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4235 }(213,a) \;\) at \(\;a = \) e.g. 2