Properties

Label 4235.3037
Modulus $4235$
Conductor $4235$
Order $44$
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(44))
 
M = H._module
 
chi = DirichletCharacter(H, M([11,22,36]))
 
pari: [g,chi] = znchar(Mod(3037,4235))
 

Basic properties

Modulus: \(4235\)
Conductor: \(4235\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
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.ce

\(\chi_{4235}(188,\cdot)\) \(\chi_{4235}(342,\cdot)\) \(\chi_{4235}(573,\cdot)\) \(\chi_{4235}(958,\cdot)\) \(\chi_{4235}(1112,\cdot)\) \(\chi_{4235}(1343,\cdot)\) \(\chi_{4235}(1497,\cdot)\) \(\chi_{4235}(1728,\cdot)\) \(\chi_{4235}(1882,\cdot)\) \(\chi_{4235}(2113,\cdot)\) \(\chi_{4235}(2267,\cdot)\) \(\chi_{4235}(2498,\cdot)\) \(\chi_{4235}(2652,\cdot)\) \(\chi_{4235}(2883,\cdot)\) \(\chi_{4235}(3037,\cdot)\) \(\chi_{4235}(3422,\cdot)\) \(\chi_{4235}(3653,\cdot)\) \(\chi_{4235}(3807,\cdot)\) \(\chi_{4235}(4038,\cdot)\) \(\chi_{4235}(4192,\cdot)\)

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: \(\Q(\zeta_{44})\)
Fixed field: Number field defined by a degree 44 polynomial

Values on generators

\((2542,1816,2906)\) → \((i,-1,e\left(\frac{9}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(12\)\(13\)\(16\)\(17\)
\( \chi_{ 4235 }(3037, a) \) \(1\)\(1\)\(e\left(\frac{3}{44}\right)\)\(i\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{9}{44}\right)\)\(-1\)\(e\left(\frac{17}{44}\right)\)\(e\left(\frac{39}{44}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{37}{44}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4235 }(3037,a) \;\) at \(\;a = \) e.g. 2