Properties

Label 61.6
Modulus $61$
Conductor $61$
Order $60$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(61)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([7]))
 
pari: [g,chi] = znchar(Mod(6,61))
 

Basic properties

Modulus: \(61\)
Conductor: \(61\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 61.l

\(\chi_{61}(2,\cdot)\) \(\chi_{61}(6,\cdot)\) \(\chi_{61}(7,\cdot)\) \(\chi_{61}(10,\cdot)\) \(\chi_{61}(17,\cdot)\) \(\chi_{61}(18,\cdot)\) \(\chi_{61}(26,\cdot)\) \(\chi_{61}(30,\cdot)\) \(\chi_{61}(31,\cdot)\) \(\chi_{61}(35,\cdot)\) \(\chi_{61}(43,\cdot)\) \(\chi_{61}(44,\cdot)\) \(\chi_{61}(51,\cdot)\) \(\chi_{61}(54,\cdot)\) \(\chi_{61}(55,\cdot)\) \(\chi_{61}(59,\cdot)\)

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

Values on generators

\(2\) → \(e\left(\frac{7}{60}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(-1\)\(1\)\(e\left(\frac{7}{60}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{49}{60}\right)\)\(e\left(\frac{43}{60}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{41}{60}\right)\)\(-i\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 61 }(6,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{61}(6,\cdot)) = \sum_{r\in \Z/61\Z} \chi_{61}(6,r) e\left(\frac{2r}{61}\right) = 3.3746873492+-7.0435420986i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 61 }(6,·),\chi_{ 61 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{61}(6,\cdot),\chi_{61}(1,\cdot)) = \sum_{r\in \Z/61\Z} \chi_{61}(6,r) \chi_{61}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 61 }(6,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{61}(6,·)) = \sum_{r \in \Z/61\Z} \chi_{61}(6,r) e\left(\frac{1 r + 2 r^{-1}}{61}\right) = -3.4387920001+8.9583594352i \)