Properties

Modulus 41
Conductor 41
Order 40
Real no
Primitive yes
Minimal yes
Parity odd
Orbit label 41.h

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([7]))
 
pari: [g,chi] = znchar(Mod(29,41))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 41
Conductor = 41
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 40
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = yes
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = odd
Orbit label = 41.h
Orbit index = 8

Galois orbit

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

\(\chi_{41}(6,\cdot)\) \(\chi_{41}(7,\cdot)\) \(\chi_{41}(11,\cdot)\) \(\chi_{41}(12,\cdot)\) \(\chi_{41}(13,\cdot)\) \(\chi_{41}(15,\cdot)\) \(\chi_{41}(17,\cdot)\) \(\chi_{41}(19,\cdot)\) \(\chi_{41}(22,\cdot)\) \(\chi_{41}(24,\cdot)\) \(\chi_{41}(26,\cdot)\) \(\chi_{41}(28,\cdot)\) \(\chi_{41}(29,\cdot)\) \(\chi_{41}(30,\cdot)\) \(\chi_{41}(34,\cdot)\) \(\chi_{41}(35,\cdot)\)

Values on generators

\(6\) → \(e\left(\frac{7}{40}\right)\)

Values

-11234567891011
\(-1\)\(1\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{7}{40}\right)\)\(e\left(\frac{33}{40}\right)\)\(e\left(\frac{13}{20}\right)\)\(i\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{21}{40}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{40})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 41 }(29,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{41}(29,\cdot)) = \sum_{r\in \Z/41\Z} \chi_{41}(29,r) e\left(\frac{2r}{41}\right) = 1.4741392179+6.2311245828i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 41 }(29,·),\chi_{ 41 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{41}(29,\cdot),\chi_{41}(1,\cdot)) = \sum_{r\in \Z/41\Z} \chi_{41}(29,r) \chi_{41}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 41 }(29,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{41}(29,·)) = \sum_{r \in \Z/41\Z} \chi_{41}(29,r) e\left(\frac{1 r + 2 r^{-1}}{41}\right) = -1.8357110379+-0.2907480653i \)