Properties

Modulus 87
Conductor 29
Order 7
Real no
Primitive no
Minimal yes
Parity even
Orbit label 87.g

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(87)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,3]))
 
pari: [g,chi] = znchar(Mod(7,87))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 87
Conductor = 29
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 7
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = no
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = even
Orbit label = 87.g
Orbit index = 7

Galois orbit

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

\(\chi_{87}(7,\cdot)\) \(\chi_{87}(16,\cdot)\) \(\chi_{87}(25,\cdot)\) \(\chi_{87}(49,\cdot)\) \(\chi_{87}(52,\cdot)\) \(\chi_{87}(82,\cdot)\)

Values on generators

\((59,31)\) → \((1,e\left(\frac{3}{7}\right))\)

Values

-11245781011131416
\(1\)\(1\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{5}{7}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{7})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 87 }(7,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{87}(7,\cdot)) = \sum_{r\in \Z/87\Z} \chi_{87}(7,r) e\left(\frac{2r}{87}\right) = -4.8427624755+2.3553453261i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 87 }(7,·),\chi_{ 87 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{87}(7,\cdot),\chi_{87}(1,\cdot)) = \sum_{r\in \Z/87\Z} \chi_{87}(7,r) \chi_{87}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 87 }(7,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{87}(7,·)) = \sum_{r \in \Z/87\Z} \chi_{87}(7,r) e\left(\frac{1 r + 2 r^{-1}}{87}\right) = 4.013006188+17.5821289032i \)