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Properties

Label 300.223
Modulus $300$
Conductor $100$
Order $20$
Real no
Primitive no
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(300, base_ring=CyclotomicField(20)) M = H._module chi = DirichletCharacter(H, M([10,0,11]))
 
Copy content pari:[g,chi] = znchar(Mod(223,300))
 

Basic properties

Modulus: \(300\)
Conductor: \(100\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(20\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{100}(23,\cdot)\)
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 300.w

\(\chi_{300}(67,\cdot)\) \(\chi_{300}(103,\cdot)\) \(\chi_{300}(127,\cdot)\) \(\chi_{300}(163,\cdot)\) \(\chi_{300}(187,\cdot)\) \(\chi_{300}(223,\cdot)\) \(\chi_{300}(247,\cdot)\) \(\chi_{300}(283,\cdot)\)

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: \(\Q(\zeta_{20})\)
Fixed field: \(\Q(\zeta_{100})^+\)

Values on generators

\((151,101,277)\) → \((-1,1,e\left(\frac{11}{20}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 300 }(223, a) \) \(1\)\(1\)\(i\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{1}{5}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 300 }(223,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 300 }(223,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 300 }(223,·),\chi_{ 300 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 300 }(223,·)) \;\) at \(\; a,b = \) e.g. 1,2