Properties

Label 4235.1224
Modulus $4235$
Conductor $4235$
Order $110$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4235, base_ring=CyclotomicField(110))
 
M = H._module
 
chi = DirichletCharacter(H, M([55,55,8]))
 
pari: [g,chi] = znchar(Mod(1224,4235))
 

Basic properties

Modulus: \(4235\)
Conductor: \(4235\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(110\)
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.cs

\(\chi_{4235}(69,\cdot)\) \(\chi_{4235}(104,\cdot)\) \(\chi_{4235}(174,\cdot)\) \(\chi_{4235}(279,\cdot)\) \(\chi_{4235}(454,\cdot)\) \(\chi_{4235}(489,\cdot)\) \(\chi_{4235}(559,\cdot)\) \(\chi_{4235}(664,\cdot)\) \(\chi_{4235}(839,\cdot)\) \(\chi_{4235}(944,\cdot)\) \(\chi_{4235}(1224,\cdot)\) \(\chi_{4235}(1259,\cdot)\) \(\chi_{4235}(1329,\cdot)\) \(\chi_{4235}(1434,\cdot)\) \(\chi_{4235}(1609,\cdot)\) \(\chi_{4235}(1644,\cdot)\) \(\chi_{4235}(1714,\cdot)\) \(\chi_{4235}(1819,\cdot)\) \(\chi_{4235}(1994,\cdot)\) \(\chi_{4235}(2029,\cdot)\) \(\chi_{4235}(2099,\cdot)\) \(\chi_{4235}(2204,\cdot)\) \(\chi_{4235}(2379,\cdot)\) \(\chi_{4235}(2414,\cdot)\) \(\chi_{4235}(2484,\cdot)\) \(\chi_{4235}(2589,\cdot)\) \(\chi_{4235}(2764,\cdot)\) \(\chi_{4235}(2799,\cdot)\) \(\chi_{4235}(2869,\cdot)\) \(\chi_{4235}(2974,\cdot)\) ...

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: $\Q(\zeta_{55})$
Fixed field: Number field defined by a degree 110 polynomial (not computed)

Values on generators

\((2542,1816,2906)\) → \((-1,-1,e\left(\frac{4}{55}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(12\)\(13\)\(16\)\(17\)
\( \chi_{ 4235 }(1224, a) \) \(-1\)\(1\)\(e\left(\frac{63}{110}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{8}{55}\right)\)\(e\left(\frac{107}{110}\right)\)\(e\left(\frac{79}{110}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{19}{55}\right)\)\(e\left(\frac{16}{55}\right)\)\(e\left(\frac{31}{55}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4235 }(1224,a) \;\) at \(\;a = \) e.g. 2