Properties

Label 37.13
Modulus $37$
Conductor $37$
Order $36$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(37, base_ring=CyclotomicField(36))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([11]))
 
pari: [g,chi] = znchar(Mod(13,37))
 

Basic properties

Modulus: \(37\)
Conductor: \(37\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
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 37.i

\(\chi_{37}(2,\cdot)\) \(\chi_{37}(5,\cdot)\) \(\chi_{37}(13,\cdot)\) \(\chi_{37}(15,\cdot)\) \(\chi_{37}(17,\cdot)\) \(\chi_{37}(18,\cdot)\) \(\chi_{37}(19,\cdot)\) \(\chi_{37}(20,\cdot)\) \(\chi_{37}(22,\cdot)\) \(\chi_{37}(24,\cdot)\) \(\chi_{37}(32,\cdot)\) \(\chi_{37}(35,\cdot)\)

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

Related number fields

Field of values: \(\Q(\zeta_{36})\)
Fixed field: \(\Q(\zeta_{37})\)

Values on generators

\(2\) → \(e\left(\frac{11}{36}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(-1\)\(1\)\(e\left(\frac{11}{36}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{1}{36}\right)\)\(i\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{6}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 37 }(13,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{37}(13,\cdot)) = \sum_{r\in \Z/37\Z} \chi_{37}(13,r) e\left(\frac{2r}{37}\right) = 5.9534792188+1.2474314377i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 37 }(13,·),\chi_{ 37 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{37}(13,\cdot),\chi_{37}(1,\cdot)) = \sum_{r\in \Z/37\Z} \chi_{37}(13,r) \chi_{37}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 37 }(13,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{37}(13,·)) = \sum_{r \in \Z/37\Z} \chi_{37}(13,r) e\left(\frac{1 r + 2 r^{-1}}{37}\right) = 6.9029736755+-4.8335142037i \)