Properties

Label 637.23
Modulus $637$
Conductor $637$
Order $42$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(637, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([38,35]))
 
pari: [g,chi] = znchar(Mod(23,637))
 

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.bz

\(\chi_{637}(4,\cdot)\) \(\chi_{637}(23,\cdot)\) \(\chi_{637}(95,\cdot)\) \(\chi_{637}(114,\cdot)\) \(\chi_{637}(186,\cdot)\) \(\chi_{637}(205,\cdot)\) \(\chi_{637}(277,\cdot)\) \(\chi_{637}(296,\cdot)\) \(\chi_{637}(368,\cdot)\) \(\chi_{637}(387,\cdot)\) \(\chi_{637}(478,\cdot)\) \(\chi_{637}(550,\cdot)\)

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: 42.42.16423600478713504434070778628293678810006717122913176085381268066336462525553883868883157384200587461557.1

Values on generators

\((248,197)\) → \((e\left(\frac{19}{21}\right),e\left(\frac{5}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 637 }(23, a) \) \(1\)\(1\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{25}{42}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{20}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 637 }(23,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 637 }(23,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 637 }(23,·),\chi_{ 637 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 637 }(23,·)) \;\) at \(\; a,b = \) e.g. 1,2