Properties

Conductor 287
Order 8
Real No
Primitive Yes
Parity Even
Orbit Label 287.l

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[27]
 
pari: [g,chi] = znchar(Mod(27,287))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 287
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 8
Real = No
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = Yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = Even
Orbit label = 287.l
Orbit index = 12

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 ]
 

\(\chi_{287}(27,\cdot)\) \(\chi_{287}(55,\cdot)\) \(\chi_{287}(167,\cdot)\) \(\chi_{287}(202,\cdot)\)

Values on generators

\((206,211)\) → \((-1,e\left(\frac{1}{8}\right))\)

Values

-112345689101112
\(1\)\(1\)\(i\)\(e\left(\frac{3}{8}\right)\)\(-1\)\(i\)\(e\left(\frac{5}{8}\right)\)\(-i\)\(-i\)\(-1\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{7}{8}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{8})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 287 }(27,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{287}(27,\cdot)) = \sum_{r\in \Z/287\Z} \chi_{287}(27,r) e\left(\frac{2r}{287}\right) = -15.7767649735+-6.1720083417i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 287 }(27,·),\chi_{ 287 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{287}(27,\cdot),\chi_{287}(1,\cdot)) = \sum_{r\in \Z/287\Z} \chi_{287}(27,r) \chi_{287}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 287 }(27,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{287}(27,·)) = \sum_{r \in \Z/287\Z} \chi_{287}(27,r) e\left(\frac{1 r + 2 r^{-1}}{287}\right) = 6.5690571569+6.5690571569i \)