Properties

Label 4235.674
Modulus $4235$
Conductor $4235$
Order $330$
Real no
Primitive yes
Minimal yes
Parity even

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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([165,110,144]))
 
pari: [g,chi] = znchar(Mod(674,4235))
 

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.dk

\(\chi_{4235}(4,\cdot)\) \(\chi_{4235}(114,\cdot)\) \(\chi_{4235}(179,\cdot)\) \(\chi_{4235}(214,\cdot)\) \(\chi_{4235}(284,\cdot)\) \(\chi_{4235}(289,\cdot)\) \(\chi_{4235}(324,\cdot)\) \(\chi_{4235}(389,\cdot)\) \(\chi_{4235}(394,\cdot)\) \(\chi_{4235}(499,\cdot)\) \(\chi_{4235}(564,\cdot)\) \(\chi_{4235}(599,\cdot)\) \(\chi_{4235}(669,\cdot)\) \(\chi_{4235}(674,\cdot)\) \(\chi_{4235}(709,\cdot)\) \(\chi_{4235}(774,\cdot)\) \(\chi_{4235}(779,\cdot)\) \(\chi_{4235}(884,\cdot)\) \(\chi_{4235}(949,\cdot)\) \(\chi_{4235}(984,\cdot)\) \(\chi_{4235}(1054,\cdot)\) \(\chi_{4235}(1059,\cdot)\) \(\chi_{4235}(1094,\cdot)\) \(\chi_{4235}(1159,\cdot)\) \(\chi_{4235}(1164,\cdot)\) \(\chi_{4235}(1269,\cdot)\) \(\chi_{4235}(1369,\cdot)\) \(\chi_{4235}(1439,\cdot)\) \(\chi_{4235}(1444,\cdot)\) \(\chi_{4235}(1544,\cdot)\) ...

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

\((2542,1816,2906)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{24}{55}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(12\)\(13\)\(16\)\(17\)
\( \chi_{ 4235 }(674, a) \) \(1\)\(1\)\(e\left(\frac{199}{330}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{34}{165}\right)\)\(e\left(\frac{46}{55}\right)\)\(e\left(\frac{89}{110}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{29}{66}\right)\)\(e\left(\frac{63}{110}\right)\)\(e\left(\frac{68}{165}\right)\)\(e\left(\frac{71}{330}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4235 }(674,a) \;\) at \(\;a = \) e.g. 2