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Properties

Label 820.99
Modulus $820$
Conductor $820$
Order $40$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(820, base_ring=CyclotomicField(40)) M = H._module chi = DirichletCharacter(H, M([20,20,33]))
 
Copy content pari:[g,chi] = znchar(Mod(99,820))
 

Basic properties

Modulus: \(820\)
Conductor: \(820\)
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: yes
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 820.cc

\(\chi_{820}(19,\cdot)\) \(\chi_{820}(99,\cdot)\) \(\chi_{820}(179,\cdot)\) \(\chi_{820}(199,\cdot)\) \(\chi_{820}(239,\cdot)\) \(\chi_{820}(259,\cdot)\) \(\chi_{820}(299,\cdot)\) \(\chi_{820}(339,\cdot)\) \(\chi_{820}(399,\cdot)\) \(\chi_{820}(439,\cdot)\) \(\chi_{820}(479,\cdot)\) \(\chi_{820}(499,\cdot)\) \(\chi_{820}(539,\cdot)\) \(\chi_{820}(559,\cdot)\) \(\chi_{820}(639,\cdot)\) \(\chi_{820}(719,\cdot)\)

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: 40.40.83017628936247865363646717578368932999206418219130479061598801598873600000000000000000000.1

Values on generators

\((411,657,621)\) → \((-1,-1,e\left(\frac{33}{40}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 820 }(99, a) \) \(1\)\(1\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{7}{40}\right)\)\(-i\)\(e\left(\frac{39}{40}\right)\)\(e\left(\frac{3}{40}\right)\)\(e\left(\frac{29}{40}\right)\)\(e\left(\frac{37}{40}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{1}{8}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 820 }(99,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 820 }(99,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 820 }(99,·),\chi_{ 820 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 820 }(99,·)) \;\) at \(\; a,b = \) e.g. 1,2