Properties

Modulus 61
Conductor 61
Order 60
Real no
Primitive yes
Minimal yes
Parity odd
Orbit label 61.l

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([17]))
 
pari: [g,chi] = znchar(Mod(44,61))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 61
Conductor = 61
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 60
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = yes
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = odd
Orbit label = 61.l
Orbit index = 12

Galois orbit

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

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

Values on generators

\(2\) → \(e\left(\frac{17}{60}\right)\)

Values

-11234567891011
\(-1\)\(1\)\(e\left(\frac{17}{60}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{59}{60}\right)\)\(e\left(\frac{53}{60}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{31}{60}\right)\)\(i\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{60})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 61 }(44,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{61}(44,\cdot)) = \sum_{r\in \Z/61\Z} \chi_{61}(44,r) e\left(\frac{2r}{61}\right) = -3.2235210025+7.1139941205i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 61 }(44,·),\chi_{ 61 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{61}(44,\cdot),\chi_{61}(1,\cdot)) = \sum_{r\in \Z/61\Z} \chi_{61}(44,r) \chi_{61}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 61 }(44,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{61}(44,·)) = \sum_{r \in \Z/61\Z} \chi_{61}(44,r) e\left(\frac{1 r + 2 r^{-1}}{61}\right) = 10.9359227173+-8.8557356047i \)