Properties

Label 41.8
Modulus $41$
Conductor $41$
Order $20$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(41)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([19]))
 
pari: [g,chi] = znchar(Mod(8,41))
 

Basic properties

Modulus: \(41\)
Conductor: \(41\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
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 41.g

\(\chi_{41}(2,\cdot)\) \(\chi_{41}(5,\cdot)\) \(\chi_{41}(8,\cdot)\) \(\chi_{41}(20,\cdot)\) \(\chi_{41}(21,\cdot)\) \(\chi_{41}(33,\cdot)\) \(\chi_{41}(36,\cdot)\) \(\chi_{41}(39,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Values on generators

\(6\) → \(e\left(\frac{19}{20}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{7}{10}\right)\)\(i\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{1}{10}\right)\)\(-1\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{17}{20}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{20})\)
Fixed field: \(\Q(\zeta_{41})^+\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 41 }(8,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{41}(8,\cdot)) = \sum_{r\in \Z/41\Z} \chi_{41}(8,r) e\left(\frac{2r}{41}\right) = 0.2809067129+6.3969595449i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 41 }(8,·),\chi_{ 41 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{41}(8,\cdot),\chi_{41}(1,\cdot)) = \sum_{r\in \Z/41\Z} \chi_{41}(8,r) \chi_{41}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 41 }(8,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{41}(8,·)) = \sum_{r \in \Z/41\Z} \chi_{41}(8,r) e\left(\frac{1 r + 2 r^{-1}}{41}\right) = -0.2130837941+0.2932846818i \)