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Properties

Label 546.535
Modulus $546$
Conductor $91$
Order $12$
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(546, base_ring=CyclotomicField(12)) M = H._module chi = DirichletCharacter(H, M([0,2,1]))
 
Copy content pari:[g,chi] = znchar(Mod(535,546))
 

Basic properties

Modulus: \(546\)
Conductor: \(91\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(12\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{91}(80,\cdot)\)
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 546.by

\(\chi_{546}(19,\cdot)\) \(\chi_{546}(115,\cdot)\) \(\chi_{546}(397,\cdot)\) \(\chi_{546}(535,\cdot)\)

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: \(\Q(\zeta_{12})\)
Fixed field: 12.12.506240953553539690213.1

Values on generators

\((365,157,379)\) → \((1,e\left(\frac{1}{6}\right),e\left(\frac{1}{12}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 546 }(535, a) \) \(1\)\(1\)\(e\left(\frac{7}{12}\right)\)\(i\)\(e\left(\frac{1}{3}\right)\)\(i\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{7}{12}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 546 }(535,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 546 }(535,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 546 }(535,·),\chi_{ 546 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 546 }(535,·)) \;\) at \(\; a,b = \) e.g. 1,2