Properties

Label 95.91
Modulus $95$
Conductor $19$
Order $18$
Real no
Primitive no
Minimal yes
Parity odd

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([0,11]))
 
pari: [g,chi] = znchar(Mod(91,95))
 

Basic properties

Modulus: \(95\)
Conductor: \(19\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(18\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{19}(15,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 95.n

\(\chi_{95}(21,\cdot)\) \(\chi_{95}(41,\cdot)\) \(\chi_{95}(51,\cdot)\) \(\chi_{95}(71,\cdot)\) \(\chi_{95}(86,\cdot)\) \(\chi_{95}(91,\cdot)\)

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

\((77,21)\) → \((1,e\left(\frac{11}{18}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\(-1\)\(1\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{18}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{9})\)
Fixed field: \(\Q(\zeta_{19})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 95 }(91,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{95}(91,\cdot)) = \sum_{r\in \Z/95\Z} \chi_{95}(91,r) e\left(\frac{2r}{95}\right) = -1.9720366159+-3.8872961793i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 95 }(91,·),\chi_{ 95 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{95}(91,\cdot),\chi_{95}(1,\cdot)) = \sum_{r\in \Z/95\Z} \chi_{95}(91,r) \chi_{95}(1,1-r) = -3 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 95 }(91,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{95}(91,·)) = \sum_{r \in \Z/95\Z} \chi_{95}(91,r) e\left(\frac{1 r + 2 r^{-1}}{95}\right) = 6.7759998531+2.4662622539i \)