Properties

Conductor 25
Order 20
Real No
Primitive No
Parity Odd
Orbit Label 75.k

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(75)
sage: chi = H[73]
pari: [g,chi] = znchar(Mod(73,75))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 25
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 = No
sage: chi.is_odd()
pari: zncharisodd(g,chi)
Parity = Odd
Orbit label = 75.k
Orbit index = 11

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 ]

\(\chi_{75}(13,\cdot)\) \(\chi_{75}(22,\cdot)\) \(\chi_{75}(28,\cdot)\) \(\chi_{75}(37,\cdot)\) \(\chi_{75}(52,\cdot)\) \(\chi_{75}(58,\cdot)\) \(\chi_{75}(67,\cdot)\) \(\chi_{75}(73,\cdot)\)

Inducing primitive character

\(\chi_{25}(23,\cdot)\)

Values on generators

\((26,52)\) → \((1,e\left(\frac{11}{20}\right))\)

Values

-112478111314161719
\(-1\)\(1\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{1}{10}\right)\)\(-i\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{9}{10}\right)\)
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)
\( \tau_{ a }( \chi_{ 75 }(73,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{75}(73,\cdot)) = \sum_{r\in \Z/75\Z} \chi_{75}(73,r) e\left(\frac{2r}{75}\right) = 0.6266661678+-4.9605735066i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 75 }(73,·),\chi_{ 75 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{75}(73,\cdot),\chi_{75}(1,\cdot)) = \sum_{r\in \Z/75\Z} \chi_{75}(73,r) \chi_{75}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 75 }(73,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{75}(73,·)) = \sum_{r \in \Z/75\Z} \chi_{75}(73,r) e\left(\frac{1 r + 2 r^{-1}}{75}\right) = 19.6660988956+3.1148040669i \)