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Properties

Label 828.395
Modulus $828$
Conductor $276$
Order $22$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(828, base_ring=CyclotomicField(22)) M = H._module chi = DirichletCharacter(H, M([11,11,4]))
 
Copy content pari:[g,chi] = znchar(Mod(395,828))
 

Basic properties

Modulus: \(828\)
Conductor: \(276\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(22\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{276}(119,\cdot)\)
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 828.w

\(\chi_{828}(35,\cdot)\) \(\chi_{828}(71,\cdot)\) \(\chi_{828}(179,\cdot)\) \(\chi_{828}(215,\cdot)\) \(\chi_{828}(395,\cdot)\) \(\chi_{828}(611,\cdot)\) \(\chi_{828}(647,\cdot)\) \(\chi_{828}(683,\cdot)\) \(\chi_{828}(719,\cdot)\) \(\chi_{828}(791,\cdot)\)

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: \(\Q(\zeta_{11})\)
Fixed field: Number field defined by a degree 22 polynomial

Values on generators

\((415,461,649)\) → \((-1,-1,e\left(\frac{2}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 828 }(395, a) \) \(1\)\(1\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{7}{11}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 828 }(395,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 828 }(395,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 828 }(395,·),\chi_{ 828 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 828 }(395,·)) \;\) at \(\; a,b = \) e.g. 1,2