↑

Properties

Label 82.35
Modulus $82$
Conductor $41$
Order $40$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(82, base_ring=CyclotomicField(40)) M = H._module chi = DirichletCharacter(H, M([21]))
 
Copy content pari:[g,chi] = znchar(Mod(35,82))
 

Basic properties

Modulus: \(82\)
Conductor: \(41\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(40\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{41}(35,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 82.h

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

Copy content sage:chi.galois_orbit()
 
Copy content pari: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

\(47\) → \(e\left(\frac{21}{40}\right)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 82 }(35, a) \) \(-1\)\(1\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{19}{40}\right)\)\(-i\)\(e\left(\frac{23}{40}\right)\)\(e\left(\frac{11}{40}\right)\)\(e\left(\frac{17}{40}\right)\)\(e\left(\frac{13}{40}\right)\)\(e\left(\frac{29}{40}\right)\)\(e\left(\frac{7}{20}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 82 }(35,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 82 }(35,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 82 }(35,·),\chi_{ 82 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 82 }(35,·)) \;\) at \(\; a,b = \) e.g. 1,2