Properties

Conductor 65
Order 12
Real No
Primitive Yes
Parity Even
Orbit Label 65.o

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(65)
sage: chi = H[32]
pari: [g,chi] = znchar(Mod(32,65))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 65
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 12
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 = 65.o
Orbit index = 15

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 ]

\(\chi_{65}(2,\cdot)\) \(\chi_{65}(32,\cdot)\) \(\chi_{65}(33,\cdot)\) \(\chi_{65}(63,\cdot)\)

Values on generators

\((27,41)\) → \((i,e\left(\frac{5}{12}\right))\)

Values

-112346789111214
\(1\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{5}{6}\right)\)\(1\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{11}{12}\right)\)\(-i\)\(-1\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{12})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 65 }(32,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{65}(32,\cdot)) = \sum_{r\in \Z/65\Z} \chi_{65}(32,r) e\left(\frac{2r}{65}\right) = 4.0912904575+6.94703839i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 65 }(32,·),\chi_{ 65 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{65}(32,\cdot),\chi_{65}(1,\cdot)) = \sum_{r\in \Z/65\Z} \chi_{65}(32,r) \chi_{65}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 65 }(32,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{65}(32,·)) = \sum_{r \in \Z/65\Z} \chi_{65}(32,r) e\left(\frac{1 r + 2 r^{-1}}{65}\right) = 5.0455432461+-8.7391372541i \)