Properties

Label 37.7
Modulus $37$
Conductor $37$
Order $9$
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(37)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([8]))
 
pari: [g,chi] = znchar(Mod(7,37))
 

Basic properties

Modulus: \(37\)
Conductor: \(37\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(9\)
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 37.f

\(\chi_{37}(7,\cdot)\) \(\chi_{37}(9,\cdot)\) \(\chi_{37}(12,\cdot)\) \(\chi_{37}(16,\cdot)\) \(\chi_{37}(33,\cdot)\) \(\chi_{37}(34,\cdot)\)

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

\(2\) → \(e\left(\frac{8}{9}\right)\)

Values

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

Related number fields

Field of values: \(\Q(\zeta_{9})\)
Fixed field: 9.9.3512479453921.1

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 37 }(7,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{37}(7,\cdot)) = \sum_{r\in \Z/37\Z} \chi_{37}(7,r) e\left(\frac{2r}{37}\right) = 5.8264699031+1.7470685356i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 37 }(7,·),\chi_{ 37 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{37}(7,\cdot),\chi_{37}(1,\cdot)) = \sum_{r\in \Z/37\Z} \chi_{37}(7,r) \chi_{37}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 37 }(7,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{37}(7,·)) = \sum_{r \in \Z/37\Z} \chi_{37}(7,r) e\left(\frac{1 r + 2 r^{-1}}{37}\right) = -2.1262835857+0.7739039348i \)