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Properties

Label 73.45
Modulus $73$
Conductor $73$
Order $72$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(73, base_ring=CyclotomicField(72)) M = H._module chi = DirichletCharacter(H, M([13]))
 
Copy content pari:[g,chi] = znchar(Mod(45,73))
 

Basic properties

Modulus: \(73\)
Conductor: \(73\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(72\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: yes
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 73.l

\(\chi_{73}(5,\cdot)\) \(\chi_{73}(11,\cdot)\) \(\chi_{73}(13,\cdot)\) \(\chi_{73}(14,\cdot)\) \(\chi_{73}(15,\cdot)\) \(\chi_{73}(20,\cdot)\) \(\chi_{73}(26,\cdot)\) \(\chi_{73}(28,\cdot)\) \(\chi_{73}(29,\cdot)\) \(\chi_{73}(31,\cdot)\) \(\chi_{73}(33,\cdot)\) \(\chi_{73}(34,\cdot)\) \(\chi_{73}(39,\cdot)\) \(\chi_{73}(40,\cdot)\) \(\chi_{73}(42,\cdot)\) \(\chi_{73}(44,\cdot)\) \(\chi_{73}(45,\cdot)\) \(\chi_{73}(47,\cdot)\) \(\chi_{73}(53,\cdot)\) \(\chi_{73}(58,\cdot)\) \(\chi_{73}(59,\cdot)\) \(\chi_{73}(60,\cdot)\) \(\chi_{73}(62,\cdot)\) \(\chi_{73}(68,\cdot)\)

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

Related number fields

Field of values: $\Q(\zeta_{72})$
Fixed field: Number field defined by a degree 72 polynomial

Values on generators

\(5\) → \(e\left(\frac{13}{72}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 73 }(45, a) \) \(-1\)\(1\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{13}{72}\right)\)\(e\left(\frac{19}{36}\right)\)\(e\left(\frac{23}{24}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{67}{72}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 73 }(45,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 73 }(45,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 73 }(45,·),\chi_{ 73 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 73 }(45,·)) \;\) at \(\; a,b = \) e.g. 1,2