Properties

Conductor 95
Order 36
Real No
Primitive Yes
Parity Even
Orbit Label 95.r

Related objects

Learn more about

Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(95)
sage: chi = H[22]
pari: [g,chi] = znchar(Mod(22,95))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
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
sage: chi.is_odd()
pari: zncharisodd(g,chi)
Parity = Even
Orbit label = 95.r
Orbit index = 18

Galois orbit

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

\(\chi_{95}(2,\cdot)\) \(\chi_{95}(3,\cdot)\) \(\chi_{95}(13,\cdot)\) \(\chi_{95}(22,\cdot)\) \(\chi_{95}(32,\cdot)\) \(\chi_{95}(33,\cdot)\) \(\chi_{95}(48,\cdot)\) \(\chi_{95}(52,\cdot)\) \(\chi_{95}(53,\cdot)\) \(\chi_{95}(67,\cdot)\) \(\chi_{95}(72,\cdot)\) \(\chi_{95}(78,\cdot)\)

Values on generators

\((77,21)\) → \((i,e\left(\frac{13}{18}\right))\)

Values

-112346789111213
\(1\)\(1\)\(e\left(\frac{35}{36}\right)\)\(e\left(\frac{5}{36}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{13}{36}\right)\)
value at  e.g. 2

Related number fields

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

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 95 }(22,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{95}(22,\cdot)) = \sum_{r\in \Z/95\Z} \chi_{95}(22,r) e\left(\frac{2r}{95}\right) = 9.0526085769+3.6125168448i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 95 }(22,·),\chi_{ 95 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{95}(22,\cdot),\chi_{95}(1,\cdot)) = \sum_{r\in \Z/95\Z} \chi_{95}(22,r) \chi_{95}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 95 }(22,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{95}(22,·)) = \sum_{r \in \Z/95\Z} \chi_{95}(22,r) e\left(\frac{1 r + 2 r^{-1}}{95}\right) = 3.019190188+-0.2641449145i \)