Properties

Modulus 82
Conductor 41
Order 40
Real no
Primitive no
Minimal yes
Parity odd
Orbit label 82.h

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(82)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([31]))
 
pari: [g,chi] = znchar(Mod(13,82))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 82
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 = no
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = odd
Orbit label = 82.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 ]
 

\(\chi_{82}(7,\cdot)\) \(\chi_{82}(11,\cdot)\) \(\chi_{82}(13,\cdot)\) \(\chi_{82}(15,\cdot)\) \(\chi_{82}(17,\cdot)\) \(\chi_{82}(19,\cdot)\) \(\chi_{82}(29,\cdot)\) \(\chi_{82}(35,\cdot)\) \(\chi_{82}(47,\cdot)\) \(\chi_{82}(53,\cdot)\) \(\chi_{82}(63,\cdot)\) \(\chi_{82}(65,\cdot)\) \(\chi_{82}(67,\cdot)\) \(\chi_{82}(69,\cdot)\) \(\chi_{82}(71,\cdot)\) \(\chi_{82}(75,\cdot)\)

Values on generators

\(47\) → \(e\left(\frac{31}{40}\right)\)

Values

-113579111315171921
\(-1\)\(1\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{9}{40}\right)\)\(i\)\(e\left(\frac{13}{40}\right)\)\(e\left(\frac{1}{40}\right)\)\(e\left(\frac{27}{40}\right)\)\(e\left(\frac{23}{40}\right)\)\(e\left(\frac{39}{40}\right)\)\(e\left(\frac{17}{20}\right)\)
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)
 
\( \tau_{ a }( \chi_{ 82 }(13,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{82}(13,\cdot)) = \sum_{r\in \Z/82\Z} \chi_{82}(13,r) e\left(\frac{r}{41}\right) = -6.3107218566+-1.0838771372i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 82 }(13,·),\chi_{ 82 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{82}(13,\cdot),\chi_{82}(1,\cdot)) = \sum_{r\in \Z/82\Z} \chi_{82}(13,r) \chi_{82}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 82 }(13,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{82}(13,·)) = \sum_{r \in \Z/82\Z} \chi_{82}(13,r) e\left(\frac{1 r + 2 r^{-1}}{82}\right) = 0.9793256044+-1.9220347195i \)