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Properties

Label 155.111
Modulus $155$
Conductor $31$
Order $15$
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(155, base_ring=CyclotomicField(30)) M = H._module chi = DirichletCharacter(H, M([0,26]))
 
Copy content pari:[g,chi] = znchar(Mod(111,155))
 

Basic properties

Modulus: \(155\)
Conductor: \(31\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(15\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{31}(18,\cdot)\)
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 155.q

\(\chi_{155}(41,\cdot)\) \(\chi_{155}(51,\cdot)\) \(\chi_{155}(71,\cdot)\) \(\chi_{155}(76,\cdot)\) \(\chi_{155}(81,\cdot)\) \(\chi_{155}(111,\cdot)\) \(\chi_{155}(121,\cdot)\) \(\chi_{155}(131,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: Number field defined by a degree 15 polynomial

Values on generators

\((32,96)\) → \((1,e\left(\frac{13}{15}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 155 }(111, a) \) \(1\)\(1\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{8}{15}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 155 }(111,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 155 }(111,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 155 }(111,·),\chi_{ 155 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 155 }(111,·)) \;\) at \(\; a,b = \) e.g. 1,2