Properties

Label 127.88
Modulus $127$
Conductor $127$
Order $63$
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(127, base_ring=CyclotomicField(126))
 
M = H._module
 
chi = DirichletCharacter(H, M([32]))
 
pari: [g,chi] = znchar(Mod(88,127))
 

Basic properties

Modulus: \(127\)
Conductor: \(127\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(63\)
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 127.k

\(\chi_{127}(9,\cdot)\) \(\chi_{127}(11,\cdot)\) \(\chi_{127}(13,\cdot)\) \(\chi_{127}(15,\cdot)\) \(\chi_{127}(17,\cdot)\) \(\chi_{127}(18,\cdot)\) \(\chi_{127}(21,\cdot)\) \(\chi_{127}(26,\cdot)\) \(\chi_{127}(30,\cdot)\) \(\chi_{127}(31,\cdot)\) \(\chi_{127}(34,\cdot)\) \(\chi_{127}(35,\cdot)\) \(\chi_{127}(36,\cdot)\) \(\chi_{127}(41,\cdot)\) \(\chi_{127}(42,\cdot)\) \(\chi_{127}(44,\cdot)\) \(\chi_{127}(49,\cdot)\) \(\chi_{127}(60,\cdot)\) \(\chi_{127}(62,\cdot)\) \(\chi_{127}(69,\cdot)\) \(\chi_{127}(70,\cdot)\) \(\chi_{127}(71,\cdot)\) \(\chi_{127}(72,\cdot)\) \(\chi_{127}(74,\cdot)\) \(\chi_{127}(79,\cdot)\) \(\chi_{127}(81,\cdot)\) \(\chi_{127}(82,\cdot)\) \(\chi_{127}(84,\cdot)\) \(\chi_{127}(88,\cdot)\) \(\chi_{127}(98,\cdot)\) ...

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: $\Q(\zeta_{63})$
Fixed field: Number field defined by a degree 63 polynomial

Values on generators

\(3\) → \(e\left(\frac{16}{63}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 127 }(88, a) \) \(1\)\(1\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{16}{63}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{34}{63}\right)\)\(e\left(\frac{13}{63}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{32}{63}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{17}{63}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 127 }(88,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 127 }(88,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 127 }(88,·),\chi_{ 127 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 127 }(88,·)) \;\) at \(\; a,b = \) e.g. 1,2