Properties

Label 4235.1229
Modulus $4235$
Conductor $4235$
Order $330$
Real no
Primitive yes
Minimal yes
Parity odd

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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([165,220,249]))
 
pari: [g,chi] = znchar(Mod(1229,4235))
 

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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4235.di

\(\chi_{4235}(39,\cdot)\) \(\chi_{4235}(74,\cdot)\) \(\chi_{4235}(79,\cdot)\) \(\chi_{4235}(149,\cdot)\) \(\chi_{4235}(184,\cdot)\) \(\chi_{4235}(249,\cdot)\) \(\chi_{4235}(359,\cdot)\) \(\chi_{4235}(424,\cdot)\) \(\chi_{4235}(459,\cdot)\) \(\chi_{4235}(464,\cdot)\) \(\chi_{4235}(534,\cdot)\) \(\chi_{4235}(569,\cdot)\) \(\chi_{4235}(634,\cdot)\) \(\chi_{4235}(739,\cdot)\) \(\chi_{4235}(744,\cdot)\) \(\chi_{4235}(809,\cdot)\) \(\chi_{4235}(849,\cdot)\) \(\chi_{4235}(919,\cdot)\) \(\chi_{4235}(954,\cdot)\) \(\chi_{4235}(1019,\cdot)\) \(\chi_{4235}(1124,\cdot)\) \(\chi_{4235}(1194,\cdot)\) \(\chi_{4235}(1229,\cdot)\) \(\chi_{4235}(1234,\cdot)\) \(\chi_{4235}(1339,\cdot)\) \(\chi_{4235}(1404,\cdot)\) \(\chi_{4235}(1509,\cdot)\) \(\chi_{4235}(1514,\cdot)\) \(\chi_{4235}(1579,\cdot)\) \(\chi_{4235}(1614,\cdot)\) ...

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

\((2542,1816,2906)\) → \((-1,e\left(\frac{2}{3}\right),e\left(\frac{83}{110}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(12\)\(13\)\(16\)\(17\)
\( \chi_{ 4235 }(1229, a) \) \(-1\)\(1\)\(e\left(\frac{97}{165}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{29}{165}\right)\)\(e\left(\frac{17}{110}\right)\)\(e\left(\frac{42}{55}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{49}{66}\right)\)\(e\left(\frac{39}{55}\right)\)\(e\left(\frac{58}{165}\right)\)\(e\left(\frac{23}{165}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4235 }(1229,a) \;\) at \(\;a = \) e.g. 2