Properties

Label 4235.758
Modulus $4235$
Conductor $4235$
Order $132$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4235, base_ring=CyclotomicField(132))
 
M = H._module
 
chi = DirichletCharacter(H, M([99,44,6]))
 
pari: [g,chi] = znchar(Mod(758,4235))
 

Basic properties

Modulus: \(4235\)
Conductor: \(4235\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(132\)
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.db

\(\chi_{4235}(32,\cdot)\) \(\chi_{4235}(142,\cdot)\) \(\chi_{4235}(263,\cdot)\) \(\chi_{4235}(373,\cdot)\) \(\chi_{4235}(417,\cdot)\) \(\chi_{4235}(527,\cdot)\) \(\chi_{4235}(648,\cdot)\) \(\chi_{4235}(758,\cdot)\) \(\chi_{4235}(802,\cdot)\) \(\chi_{4235}(912,\cdot)\) \(\chi_{4235}(1033,\cdot)\) \(\chi_{4235}(1143,\cdot)\) \(\chi_{4235}(1187,\cdot)\) \(\chi_{4235}(1297,\cdot)\) \(\chi_{4235}(1418,\cdot)\) \(\chi_{4235}(1528,\cdot)\) \(\chi_{4235}(1682,\cdot)\) \(\chi_{4235}(1803,\cdot)\) \(\chi_{4235}(1913,\cdot)\) \(\chi_{4235}(1957,\cdot)\) \(\chi_{4235}(2067,\cdot)\) \(\chi_{4235}(2188,\cdot)\) \(\chi_{4235}(2342,\cdot)\) \(\chi_{4235}(2452,\cdot)\) \(\chi_{4235}(2573,\cdot)\) \(\chi_{4235}(2683,\cdot)\) \(\chi_{4235}(2727,\cdot)\) \(\chi_{4235}(2837,\cdot)\) \(\chi_{4235}(2958,\cdot)\) \(\chi_{4235}(3068,\cdot)\) ...

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{132})$
Fixed field: Number field defined by a degree 132 polynomial (not computed)

Values on generators

\((2542,1816,2906)\) → \((-i,e\left(\frac{1}{3}\right),e\left(\frac{1}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(12\)\(13\)\(16\)\(17\)
\( \chi_{ 4235 }(758, a) \) \(1\)\(1\)\(e\left(\frac{61}{132}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{61}{66}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{17}{44}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{67}{132}\right)\)\(e\left(\frac{37}{44}\right)\)\(e\left(\frac{28}{33}\right)\)\(e\left(\frac{41}{132}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4235 }(758,a) \;\) at \(\;a = \) e.g. 2