Properties

Label 61.25
Modulus $61$
Conductor $61$
Order $15$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(61, base_ring=CyclotomicField(30))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([22]))
 
pari: [g,chi] = znchar(Mod(25,61))
 

Basic properties

Modulus: \(61\)
Conductor: \(61\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(15\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 61.i

\(\chi_{61}(12,\cdot)\) \(\chi_{61}(15,\cdot)\) \(\chi_{61}(16,\cdot)\) \(\chi_{61}(22,\cdot)\) \(\chi_{61}(25,\cdot)\) \(\chi_{61}(42,\cdot)\) \(\chi_{61}(56,\cdot)\) \(\chi_{61}(57,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{15})\)
Fixed field: 15.15.9876832533361318095112441.1

Values on generators

\(2\) → \(e\left(\frac{11}{15}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{13}{15}\right)\)\(1\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 61 }(25,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{61}(25,\cdot)) = \sum_{r\in \Z/61\Z} \chi_{61}(25,r) e\left(\frac{2r}{61}\right) = 1.3496967484+7.6927445484i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 61 }(25,·),\chi_{ 61 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{61}(25,\cdot),\chi_{61}(1,\cdot)) = \sum_{r\in \Z/61\Z} \chi_{61}(25,r) \chi_{61}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 61 }(25,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{61}(25,·)) = \sum_{r \in \Z/61\Z} \chi_{61}(25,r) e\left(\frac{1 r + 2 r^{-1}}{61}\right) = 4.5044031098+-5.0026464655i \)