Properties

Conductor 49
Order 21
Real No
Primitive No
Parity Even
Orbit Label 98.g

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(98)
sage: chi = H[81]
pari: [g,chi] = znchar(Mod(81,98))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 49
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 21
Real = No
sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
Primitive = No
sage: chi.is_odd()
pari: zncharisodd(g,chi)
Parity = Even
Orbit label = 98.g
Orbit index = 7

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 ]

\(\chi_{98}(9,\cdot)\) \(\chi_{98}(11,\cdot)\) \(\chi_{98}(23,\cdot)\) \(\chi_{98}(25,\cdot)\) \(\chi_{98}(37,\cdot)\) \(\chi_{98}(39,\cdot)\) \(\chi_{98}(51,\cdot)\) \(\chi_{98}(53,\cdot)\) \(\chi_{98}(65,\cdot)\) \(\chi_{98}(81,\cdot)\) \(\chi_{98}(93,\cdot)\) \(\chi_{98}(95,\cdot)\)

Inducing primitive character

\(\chi_{49}(32,\cdot)\)

Values on generators

\(3\) → \(e\left(\frac{2}{21}\right)\)

Values

-1135911131517192325
\(1\)\(1\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{11}{21}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{21})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 98 }(81,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{98}(81,\cdot)) = \sum_{r\in \Z/98\Z} \chi_{98}(81,r) e\left(\frac{r}{49}\right) = 6.8407499752+1.4846345599i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 98 }(81,·),\chi_{ 98 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{98}(81,\cdot),\chi_{98}(1,\cdot)) = \sum_{r\in \Z/98\Z} \chi_{98}(81,r) \chi_{98}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 98 }(81,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{98}(81,·)) = \sum_{r \in \Z/98\Z} \chi_{98}(81,r) e\left(\frac{1 r + 2 r^{-1}}{98}\right) = -0.122692573+-1.6372186062i \)