Properties

Conductor 29
Order 4
Real No
Primitive No
Parity Odd
Orbit Label 87.e

Related objects

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Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(87)
sage: chi = H[70]
pari: [g,chi] = znchar(Mod(70,87))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 29
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 = 87.e
Orbit index = 5

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 ]

\(\chi_{87}(46,\cdot)\) \(\chi_{87}(70,\cdot)\)

Inducing primitive character

\(\chi_{29}(12,\cdot)\)

Values on generators

\((59,31)\) → \((1,i)\)

Values

-11245781011131416
\(-1\)\(1\)\(i\)\(-1\)\(-1\)\(1\)\(-i\)\(-i\)\(i\)\(-1\)\(i\)\(1\)
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)
\( \tau_{ a }( \chi_{ 87 }(70,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{87}(70,\cdot)) = \sum_{r\in \Z/87\Z} \chi_{87}(70,r) e\left(\frac{2r}{87}\right) = -1.0183751677+5.287996976i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 87 }(70,·),\chi_{ 87 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{87}(70,\cdot),\chi_{87}(1,\cdot)) = \sum_{r\in \Z/87\Z} \chi_{87}(70,r) \chi_{87}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 87 }(70,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{87}(70,·)) = \sum_{r \in \Z/87\Z} \chi_{87}(70,r) e\left(\frac{1 r + 2 r^{-1}}{87}\right) = -4.343806656+4.343806656i \)