Properties

Modulus 95
Conductor 95
Order 36
Real no
Primitive yes
Minimal yes
Parity odd
Orbit label 95.q

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(95)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([9,16]))
 
pari: [g,chi] = znchar(Mod(47,95))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 95
Conductor = 95
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 36
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = yes
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = odd
Orbit label = 95.q
Orbit index = 17

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 ]
 

\(\chi_{95}(17,\cdot)\) \(\chi_{95}(23,\cdot)\) \(\chi_{95}(28,\cdot)\) \(\chi_{95}(42,\cdot)\) \(\chi_{95}(43,\cdot)\) \(\chi_{95}(47,\cdot)\) \(\chi_{95}(62,\cdot)\) \(\chi_{95}(63,\cdot)\) \(\chi_{95}(73,\cdot)\) \(\chi_{95}(82,\cdot)\) \(\chi_{95}(92,\cdot)\) \(\chi_{95}(93,\cdot)\)

Values on generators

\((77,21)\) → \((i,e\left(\frac{4}{9}\right))\)

Values

-112346789111213
\(-1\)\(1\)\(e\left(\frac{25}{36}\right)\)\(e\left(\frac{19}{36}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{35}{36}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{36})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 95 }(47,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{95}(47,\cdot)) = \sum_{r\in \Z/95\Z} \chi_{95}(47,r) e\left(\frac{2r}{95}\right) = 2.9109439349+9.3019570741i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 95 }(47,·),\chi_{ 95 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{95}(47,\cdot),\chi_{95}(1,\cdot)) = \sum_{r\in \Z/95\Z} \chi_{95}(47,r) \chi_{95}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 95 }(47,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{95}(47,·)) = \sum_{r \in \Z/95\Z} \chi_{95}(47,r) e\left(\frac{1 r + 2 r^{-1}}{95}\right) = 4.8771883581+3.4150440536i \)