Properties

Conductor 5
Order 4
Real No
Primitive No
Parity Odd
Orbit Label 75.f

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[43]
pari: [g,chi] = znchar(Mod(43,75))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 5
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 4
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.f
Orbit index = 6

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 ]

\(\chi_{75}(7,\cdot)\) \(\chi_{75}(43,\cdot)\)

Inducing primitive character

\(\chi_{5}(3,\cdot)\)

Values on generators

\((26,52)\) → \((1,-i)\)

Values

-112478111314161719
\(-1\)\(1\)\(-i\)\(-1\)\(-i\)\(i\)\(1\)\(i\)\(-1\)\(1\)\(-i\)\(-1\)
value at  e.g. 2

Related number fields

Field of values \(\Q(i)\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 75 }(43,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{75}(43,\cdot)) = \sum_{r\in \Z/75\Z} \chi_{75}(43,r) e\left(\frac{2r}{75}\right) = -0.0 \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 75 }(43,·),\chi_{ 75 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{75}(43,\cdot),\chi_{75}(1,\cdot)) = \sum_{r\in \Z/75\Z} \chi_{75}(43,r) \chi_{75}(1,1-r) = -5 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 75 }(43,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{75}(43,·)) = \sum_{r \in \Z/75\Z} \chi_{75}(43,r) e\left(\frac{1 r + 2 r^{-1}}{75}\right) = -0.0 \)