Properties

Conductor 55
Order 20
Real No
Primitive Yes
Parity Even
Orbit Label 55.l

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(55)
sage: chi = H[17]
pari: [g,chi] = znchar(Mod(17,55))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 55
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 20
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 = 55.l
Orbit index = 12

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 ]

\(\chi_{55}(2,\cdot)\) \(\chi_{55}(7,\cdot)\) \(\chi_{55}(8,\cdot)\) \(\chi_{55}(13,\cdot)\) \(\chi_{55}(17,\cdot)\) \(\chi_{55}(18,\cdot)\) \(\chi_{55}(28,\cdot)\) \(\chi_{55}(52,\cdot)\)

Values on generators

\((12,46)\) → \((i,e\left(\frac{9}{10}\right))\)

Values

-112346789121314
\(1\)\(1\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{9}{10}\right)\)\(i\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{7}{10}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{20})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 55 }(17,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{55}(17,\cdot)) = \sum_{r\in \Z/55\Z} \chi_{55}(17,r) e\left(\frac{2r}{55}\right) = 7.4058795486+-0.3910858118i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 55 }(17,·),\chi_{ 55 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{55}(17,\cdot),\chi_{55}(1,\cdot)) = \sum_{r\in \Z/55\Z} \chi_{55}(17,r) \chi_{55}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 55 }(17,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{55}(17,·)) = \sum_{r \in \Z/55\Z} \chi_{55}(17,r) e\left(\frac{1 r + 2 r^{-1}}{55}\right) = 4.7556526324+2.4231260452i \)