Properties

Label 83.24
Modulus $83$
Conductor $83$
Order $82$
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(83)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([75]))
 
pari: [g,chi] = znchar(Mod(24,83))
 

Basic properties

Modulus: \(83\)
Conductor: \(83\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(82\)
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 83.d

\(\chi_{83}(2,\cdot)\) \(\chi_{83}(5,\cdot)\) \(\chi_{83}(6,\cdot)\) \(\chi_{83}(8,\cdot)\) \(\chi_{83}(13,\cdot)\) \(\chi_{83}(14,\cdot)\) \(\chi_{83}(15,\cdot)\) \(\chi_{83}(18,\cdot)\) \(\chi_{83}(19,\cdot)\) \(\chi_{83}(20,\cdot)\) \(\chi_{83}(22,\cdot)\) \(\chi_{83}(24,\cdot)\) \(\chi_{83}(32,\cdot)\) \(\chi_{83}(34,\cdot)\) \(\chi_{83}(35,\cdot)\) \(\chi_{83}(39,\cdot)\) \(\chi_{83}(42,\cdot)\) \(\chi_{83}(43,\cdot)\) \(\chi_{83}(45,\cdot)\) \(\chi_{83}(46,\cdot)\) \(\chi_{83}(47,\cdot)\) \(\chi_{83}(50,\cdot)\) \(\chi_{83}(52,\cdot)\) \(\chi_{83}(53,\cdot)\) \(\chi_{83}(54,\cdot)\) \(\chi_{83}(55,\cdot)\) \(\chi_{83}(56,\cdot)\) \(\chi_{83}(57,\cdot)\) \(\chi_{83}(58,\cdot)\) \(\chi_{83}(60,\cdot)\) ...

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

\(2\) → \(e\left(\frac{75}{82}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(-1\)\(1\)\(e\left(\frac{75}{82}\right)\)\(e\left(\frac{35}{41}\right)\)\(e\left(\frac{34}{41}\right)\)\(e\left(\frac{57}{82}\right)\)\(e\left(\frac{63}{82}\right)\)\(e\left(\frac{13}{41}\right)\)\(e\left(\frac{61}{82}\right)\)\(e\left(\frac{29}{41}\right)\)\(e\left(\frac{25}{41}\right)\)\(e\left(\frac{39}{41}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{41})$
Fixed field: Number field defined by a degree 82 polynomial

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 83 }(24,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{83}(24,\cdot)) = \sum_{r\in \Z/83\Z} \chi_{83}(24,r) e\left(\frac{2r}{83}\right) = 8.921061098+-1.8478822703i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 83 }(24,·),\chi_{ 83 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{83}(24,\cdot),\chi_{83}(1,\cdot)) = \sum_{r\in \Z/83\Z} \chi_{83}(24,r) \chi_{83}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 83 }(24,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{83}(24,·)) = \sum_{r \in \Z/83\Z} \chi_{83}(24,r) e\left(\frac{1 r + 2 r^{-1}}{83}\right) = -1.3351956932+-4.8587056214i \)