Properties

Label 4235.233
Modulus $4235$
Conductor $385$
Order $60$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4235, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([45,20,6]))
 
pari: [g,chi] = znchar(Mod(233,4235))
 

Basic properties

Modulus: \(4235\)
Conductor: \(385\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{385}(233,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4235.cj

\(\chi_{4235}(233,\cdot)\) \(\chi_{4235}(282,\cdot)\) \(\chi_{4235}(403,\cdot)\) \(\chi_{4235}(457,\cdot)\) \(\chi_{4235}(578,\cdot)\) \(\chi_{4235}(723,\cdot)\) \(\chi_{4235}(1927,\cdot)\) \(\chi_{4235}(2048,\cdot)\) \(\chi_{4235}(2097,\cdot)\) \(\chi_{4235}(2272,\cdot)\) \(\chi_{4235}(2417,\cdot)\) \(\chi_{4235}(2538,\cdot)\) \(\chi_{4235}(2823,\cdot)\) \(\chi_{4235}(2998,\cdot)\) \(\chi_{4235}(3742,\cdot)\) \(\chi_{4235}(4232,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((2542,1816,2906)\) → \((-i,e\left(\frac{1}{3}\right),e\left(\frac{1}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(12\)\(13\)\(16\)\(17\)
\( \chi_{ 4235 }(233, a) \) \(1\)\(1\)\(e\left(\frac{31}{60}\right)\)\(e\left(\frac{23}{60}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{59}{60}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4235 }(233,a) \;\) at \(\;a = \) e.g. 2