Properties

Label 165.74
Modulus $165$
Conductor $165$
Order $10$
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(165, base_ring=CyclotomicField(10))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([5,5,3]))
 
pari: [g,chi] = znchar(Mod(74,165))
 

Basic properties

Modulus: \(165\)
Conductor: \(165\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(10\)
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 165.r

\(\chi_{165}(29,\cdot)\) \(\chi_{165}(74,\cdot)\) \(\chi_{165}(134,\cdot)\) \(\chi_{165}(149,\cdot)\)

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

\((56,67,46)\) → \((-1,-1,e\left(\frac{3}{10}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(13\)\(14\)\(16\)\(17\)\(19\)\(23\)
\(1\)\(1\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{9}{10}\right)\)\(1\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{5})\)
Fixed field: 10.10.1790566527853125.1

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 165 }(74,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{165}(74,\cdot)) = \sum_{r\in \Z/165\Z} \chi_{165}(74,r) e\left(\frac{2r}{165}\right) = -6.8099820167+-10.8914712015i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 165 }(74,·),\chi_{ 165 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{165}(74,\cdot),\chi_{165}(1,\cdot)) = \sum_{r\in \Z/165\Z} \chi_{165}(74,r) \chi_{165}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 165 }(74,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{165}(74,·)) = \sum_{r \in \Z/165\Z} \chi_{165}(74,r) e\left(\frac{1 r + 2 r^{-1}}{165}\right) = -0.0 \)