Properties

Label 4235.1459
Modulus $4235$
Conductor $4235$
Order $330$
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(330))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([165,55,21]))
 
pari: [g,chi] = znchar(Mod(1459,4235))
 

Basic properties

Modulus: \(4235\)
Conductor: \(4235\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(330\)
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.dj

\(\chi_{4235}(19,\cdot)\) \(\chi_{4235}(24,\cdot)\) \(\chi_{4235}(129,\cdot)\) \(\chi_{4235}(194,\cdot)\) \(\chi_{4235}(299,\cdot)\) \(\chi_{4235}(304,\cdot)\) \(\chi_{4235}(369,\cdot)\) \(\chi_{4235}(404,\cdot)\) \(\chi_{4235}(409,\cdot)\) \(\chi_{4235}(479,\cdot)\) \(\chi_{4235}(514,\cdot)\) \(\chi_{4235}(579,\cdot)\) \(\chi_{4235}(684,\cdot)\) \(\chi_{4235}(689,\cdot)\) \(\chi_{4235}(754,\cdot)\) \(\chi_{4235}(789,\cdot)\) \(\chi_{4235}(794,\cdot)\) \(\chi_{4235}(864,\cdot)\) \(\chi_{4235}(899,\cdot)\) \(\chi_{4235}(964,\cdot)\) \(\chi_{4235}(1069,\cdot)\) \(\chi_{4235}(1074,\cdot)\) \(\chi_{4235}(1139,\cdot)\) \(\chi_{4235}(1174,\cdot)\) \(\chi_{4235}(1179,\cdot)\) \(\chi_{4235}(1249,\cdot)\) \(\chi_{4235}(1284,\cdot)\) \(\chi_{4235}(1349,\cdot)\) \(\chi_{4235}(1454,\cdot)\) \(\chi_{4235}(1459,\cdot)\) ...

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: $\Q(\zeta_{165})$
Fixed field: Number field defined by a degree 330 polynomial (not computed)

Values on generators

\((2542,1816,2906)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{7}{110}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(12\)\(13\)\(16\)\(17\)
\(1\)\(1\)\(e\left(\frac{148}{165}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{131}{165}\right)\)\(e\left(\frac{9}{55}\right)\)\(e\left(\frac{38}{55}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{2}{33}\right)\)\(e\left(\frac{47}{110}\right)\)\(e\left(\frac{97}{165}\right)\)\(e\left(\frac{259}{330}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4235 }(1459,a) \;\) at \(\;a = \) e.g. 2