Properties

Label 4235.3019
Modulus $4235$
Conductor $4235$
Order $330$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4235, base_ring=CyclotomicField(330))
 
M = H._module
 
chi = DirichletCharacter(H, M([165,110,102]))
 
pari: [g,chi] = znchar(Mod(3019,4235))
 

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

Related number fields

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

Values on generators

\((2542,1816,2906)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{17}{55}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(12\)\(13\)\(16\)\(17\)
\( \chi_{ 4235 }(3019, a) \) \(1\)\(1\)\(e\left(\frac{157}{330}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{157}{165}\right)\)\(e\left(\frac{28}{55}\right)\)\(e\left(\frac{47}{110}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{65}{66}\right)\)\(e\left(\frac{79}{110}\right)\)\(e\left(\frac{149}{165}\right)\)\(e\left(\frac{323}{330}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4235 }(3019,a) \;\) at \(\;a = \) e.g. 2