Properties

Conductor 287
Order 15
Real no
Primitive yes
Minimal yes
Parity even
Orbit label 287.s

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(287)
 
sage: chi = H[256]
 
pari: [g,chi] = znchar(Mod(256,287))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 287
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 15
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = yes
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = even
Orbit label = 287.s
Orbit index = 19

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 ]
 

\(\chi_{287}(16,\cdot)\) \(\chi_{287}(18,\cdot)\) \(\chi_{287}(37,\cdot)\) \(\chi_{287}(51,\cdot)\) \(\chi_{287}(100,\cdot)\) \(\chi_{287}(221,\cdot)\) \(\chi_{287}(242,\cdot)\) \(\chi_{287}(256,\cdot)\)

Values on generators

\((206,211)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{1}{5}\right))\)

Values

-112345689101112
\(1\)\(1\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{11}{15}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{15})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 287 }(256,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{287}(256,\cdot)) = \sum_{r\in \Z/287\Z} \chi_{287}(256,r) e\left(\frac{2r}{287}\right) = -16.2693601789+-4.7231260167i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 287 }(256,·),\chi_{ 287 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{287}(256,\cdot),\chi_{287}(1,\cdot)) = \sum_{r\in \Z/287\Z} \chi_{287}(256,r) \chi_{287}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 287 }(256,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{287}(256,·)) = \sum_{r \in \Z/287\Z} \chi_{287}(256,r) e\left(\frac{1 r + 2 r^{-1}}{287}\right) = -1.6062768709+15.2827035641i \)