Properties

Conductor 41
Order 4
Real No
Primitive No
Parity Even
Orbit Label 287.f

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(287)
 
sage: chi = H[50]
 
pari: [g,chi] = znchar(Mod(50,287))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 41
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 = Even
Orbit label = 287.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 ]
 

\(\chi_{287}(50,\cdot)\) \(\chi_{287}(155,\cdot)\)

Inducing primitive character

\(\chi_{41}(9,\cdot)\)

Values on generators

\((206,211)\) → \((1,-i)\)

Values

-112345689101112
\(1\)\(1\)\(-1\)\(i\)\(1\)\(-1\)\(-i\)\(-1\)\(-1\)\(1\)\(i\)\(i\)
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)
 
\( \tau_{ a }( \chi_{ 287 }(50,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{287}(50,\cdot)) = \sum_{r\in \Z/287\Z} \chi_{287}(50,r) e\left(\frac{2r}{287}\right) = -6.0421693615+-2.1194785695i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 287 }(50,·),\chi_{ 287 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{287}(50,\cdot),\chi_{287}(1,\cdot)) = \sum_{r\in \Z/287\Z} \chi_{287}(50,r) \chi_{287}(1,1-r) = -5 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 287 }(50,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{287}(50,·)) = \sum_{r \in \Z/287\Z} \chi_{287}(50,r) e\left(\frac{1 r + 2 r^{-1}}{287}\right) = 7.6744345422i \)