Properties

Label 1031.24
Modulus $1031$
Conductor $1031$
Order $515$
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(1031, base_ring=CyclotomicField(1030))
 
M = H._module
 
chi = DirichletCharacter(H, M([934]))
 
pari: [g,chi] = znchar(Mod(24,1031))
 

Basic properties

Modulus: \(1031\)
Conductor: \(1031\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(515\)
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 1031.g

\(\chi_{1031}(2,\cdot)\) \(\chi_{1031}(3,\cdot)\) \(\chi_{1031}(4,\cdot)\) \(\chi_{1031}(5,\cdot)\) \(\chi_{1031}(6,\cdot)\) \(\chi_{1031}(8,\cdot)\) \(\chi_{1031}(9,\cdot)\) \(\chi_{1031}(11,\cdot)\) \(\chi_{1031}(12,\cdot)\) \(\chi_{1031}(13,\cdot)\) \(\chi_{1031}(16,\cdot)\) \(\chi_{1031}(18,\cdot)\) \(\chi_{1031}(20,\cdot)\) \(\chi_{1031}(22,\cdot)\) \(\chi_{1031}(23,\cdot)\) \(\chi_{1031}(24,\cdot)\) \(\chi_{1031}(25,\cdot)\) \(\chi_{1031}(27,\cdot)\) \(\chi_{1031}(30,\cdot)\) \(\chi_{1031}(33,\cdot)\) \(\chi_{1031}(36,\cdot)\) \(\chi_{1031}(40,\cdot)\) \(\chi_{1031}(43,\cdot)\) \(\chi_{1031}(44,\cdot)\) \(\chi_{1031}(45,\cdot)\) \(\chi_{1031}(46,\cdot)\) \(\chi_{1031}(47,\cdot)\) \(\chi_{1031}(50,\cdot)\) \(\chi_{1031}(52,\cdot)\) \(\chi_{1031}(53,\cdot)\) ...

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: $\Q(\zeta_{515})$
Fixed field: Number field defined by a degree 515 polynomial (not computed)

Values on generators

\(14\) → \(e\left(\frac{467}{515}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1031 }(24, a) \) \(1\)\(1\)\(e\left(\frac{417}{515}\right)\)\(e\left(\frac{267}{515}\right)\)\(e\left(\frac{319}{515}\right)\)\(e\left(\frac{108}{515}\right)\)\(e\left(\frac{169}{515}\right)\)\(e\left(\frac{10}{103}\right)\)\(e\left(\frac{221}{515}\right)\)\(e\left(\frac{19}{515}\right)\)\(e\left(\frac{2}{103}\right)\)\(e\left(\frac{439}{515}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1031 }(24,a) \;\) at \(\;a = \) e.g. 2