Properties

Label 637.36
Modulus $637$
Conductor $637$
Order $42$
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(637, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([12,35]))
 
pari: [g,chi] = znchar(Mod(36,637))
 

Basic properties

Modulus: \(637\)
Conductor: \(637\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
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 637.bt

\(\chi_{637}(36,\cdot)\) \(\chi_{637}(43,\cdot)\) \(\chi_{637}(127,\cdot)\) \(\chi_{637}(134,\cdot)\) \(\chi_{637}(218,\cdot)\) \(\chi_{637}(225,\cdot)\) \(\chi_{637}(309,\cdot)\) \(\chi_{637}(316,\cdot)\) \(\chi_{637}(400,\cdot)\) \(\chi_{637}(407,\cdot)\) \(\chi_{637}(498,\cdot)\) \(\chi_{637}(582,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: Number field defined by a degree 42 polynomial

Values on generators

\((248,197)\) → \((e\left(\frac{2}{7}\right),e\left(\frac{5}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 637 }(36, a) \) \(1\)\(1\)\(e\left(\frac{11}{42}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{37}{42}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{11}{42}\right)\)\(e\left(\frac{1}{7}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 637 }(36,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 637 }(36,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 637 }(36,·),\chi_{ 637 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 637 }(36,·)) \;\) at \(\; a,b = \) e.g. 1,2