Properties

Label 87.50
Modulus $87$
Conductor $87$
Order $28$
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(87, base_ring=CyclotomicField(28))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([14,17]))
 
pari: [g,chi] = znchar(Mod(50,87))
 

Basic properties

Modulus: \(87\)
Conductor: \(87\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(28\)
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 87.k

\(\chi_{87}(2,\cdot)\) \(\chi_{87}(8,\cdot)\) \(\chi_{87}(11,\cdot)\) \(\chi_{87}(14,\cdot)\) \(\chi_{87}(26,\cdot)\) \(\chi_{87}(32,\cdot)\) \(\chi_{87}(44,\cdot)\) \(\chi_{87}(47,\cdot)\) \(\chi_{87}(50,\cdot)\) \(\chi_{87}(56,\cdot)\) \(\chi_{87}(68,\cdot)\) \(\chi_{87}(77,\cdot)\)

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: \(\Q(\zeta_{28})\)
Fixed field: \(\Q(\zeta_{87})^+\)

Values on generators

\((59,31)\) → \((-1,e\left(\frac{17}{28}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(1\)\(1\)\(e\left(\frac{3}{28}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{9}{28}\right)\)\(e\left(\frac{27}{28}\right)\)\(e\left(\frac{19}{28}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{11}{28}\right)\)\(e\left(\frac{3}{7}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 87 }(50,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{87}(50,\cdot)) = \sum_{r\in \Z/87\Z} \chi_{87}(50,r) e\left(\frac{2r}{87}\right) = 8.9734974327+2.5448661703i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 87 }(50,·),\chi_{ 87 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{87}(50,\cdot),\chi_{87}(1,\cdot)) = \sum_{r\in \Z/87\Z} \chi_{87}(50,r) \chi_{87}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 87 }(50,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{87}(50,·)) = \sum_{r \in \Z/87\Z} \chi_{87}(50,r) e\left(\frac{1 r + 2 r^{-1}}{87}\right) = 0.0 \)