Properties

Label 561.2
Modulus $561$
Conductor $561$
Order $40$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(561, base_ring=CyclotomicField(40))
 
M = H._module
 
chi = DirichletCharacter(H, M([20,4,35]))
 
pari: [g,chi] = znchar(Mod(2,561))
 

Basic properties

Modulus: \(561\)
Conductor: \(561\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(40\)
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 561.bg

\(\chi_{561}(2,\cdot)\) \(\chi_{561}(8,\cdot)\) \(\chi_{561}(83,\cdot)\) \(\chi_{561}(128,\cdot)\) \(\chi_{561}(134,\cdot)\) \(\chi_{561}(161,\cdot)\) \(\chi_{561}(206,\cdot)\) \(\chi_{561}(281,\cdot)\) \(\chi_{561}(314,\cdot)\) \(\chi_{561}(332,\cdot)\) \(\chi_{561}(338,\cdot)\) \(\chi_{561}(359,\cdot)\) \(\chi_{561}(365,\cdot)\) \(\chi_{561}(491,\cdot)\) \(\chi_{561}(512,\cdot)\) \(\chi_{561}(536,\cdot)\)

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: \(\Q(\zeta_{40})\)
Fixed field: Number field defined by a degree 40 polynomial

Values on generators

\((188,409,496)\) → \((-1,e\left(\frac{1}{10}\right),e\left(\frac{7}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(13\)\(14\)\(16\)\(19\)
\( \chi_{ 561 }(2, a) \) \(1\)\(1\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{11}{40}\right)\)\(e\left(\frac{13}{40}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{7}{40}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{11}{20}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 561 }(2,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 561 }(2,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 561 }(2,·),\chi_{ 561 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 561 }(2,·)) \;\) at \(\; a,b = \) e.g. 1,2