Properties

Label 61.39
Modulus $61$
Conductor $61$
Order $30$
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([23]))
 
pari: [g,chi] = znchar(Mod(39,61))
 

Basic properties

Modulus: \(61\)
Conductor: \(61\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
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.k

\(\chi_{61}(4,\cdot)\) \(\chi_{61}(5,\cdot)\) \(\chi_{61}(19,\cdot)\) \(\chi_{61}(36,\cdot)\) \(\chi_{61}(39,\cdot)\) \(\chi_{61}(45,\cdot)\) \(\chi_{61}(46,\cdot)\) \(\chi_{61}(49,\cdot)\)

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: \(\Q(\zeta_{61})^+\)

Values on generators

\(2\) → \(e\left(\frac{23}{30}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{19}{30}\right)\)\(-1\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 61 }(39,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{61}(39,\cdot)) = \sum_{r\in \Z/61\Z} \chi_{61}(39,r) e\left(\frac{2r}{61}\right) = 0.2397067003+-7.8065703544i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 61 }(39,·),\chi_{ 61 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{61}(39,\cdot),\chi_{61}(1,\cdot)) = \sum_{r\in \Z/61\Z} \chi_{61}(39,r) \chi_{61}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 61 }(39,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{61}(39,·)) = \sum_{r \in \Z/61\Z} \chi_{61}(39,r) e\left(\frac{1 r + 2 r^{-1}}{61}\right) = 1.5757458215+-1.4188079104i \)