Properties

Label 95.63
Modulus $95$
Conductor $95$
Order $36$
Real no
Primitive yes
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([27,28]))
 
pari: [g,chi] = znchar(Mod(63,95))
 

Basic properties

Modulus: \(95\)
Conductor: \(95\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 95.q

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

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

\((77,21)\) → \((-i,e\left(\frac{7}{9}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\(-1\)\(1\)\(e\left(\frac{19}{36}\right)\)\(e\left(\frac{13}{36}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{5}{36}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{36})\)
Fixed field: 36.0.619876750267203693326033178758188478035934269428253173828125.1

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 95 }(63,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{95}(63,\cdot)) = \sum_{r\in \Z/95\Z} \chi_{95}(63,r) e\left(\frac{2r}{95}\right) = -9.5125913482+2.1238186934i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 95 }(63,·),\chi_{ 95 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{95}(63,\cdot),\chi_{95}(1,\cdot)) = \sum_{r\in \Z/95\Z} \chi_{95}(63,r) \chi_{95}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 95 }(63,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{95}(63,·)) = \sum_{r \in \Z/95\Z} \chi_{95}(63,r) e\left(\frac{1 r + 2 r^{-1}}{95}\right) = -1.8357405693+-0.160606489i \)