Properties

Conductor 287
Order 12
Real No
Primitive Yes
Parity Even
Orbit Label 287.r

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[214]
 
pari: [g,chi] = znchar(Mod(214,287))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 287
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 12
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.r
Orbit index = 18

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 ]
 

\(\chi_{287}(9,\cdot)\) \(\chi_{287}(32,\cdot)\) \(\chi_{287}(114,\cdot)\) \(\chi_{287}(214,\cdot)\)

Values on generators

\((206,211)\) → \((e\left(\frac{2}{3}\right),-i)\)

Values

-112345689101112
\(1\)\(1\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{6}\right)\)\(-i\)\(-1\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{7}{12}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{12})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 287 }(214,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{287}(214,\cdot)) = \sum_{r\in \Z/287\Z} \chi_{287}(214,r) e\left(\frac{2r}{287}\right) = 4.583912891+-16.3091306515i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 287 }(214,·),\chi_{ 287 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{287}(214,\cdot),\chi_{287}(1,\cdot)) = \sum_{r\in \Z/287\Z} \chi_{287}(214,r) \chi_{287}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 287 }(214,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{287}(214,·)) = \sum_{r \in \Z/287\Z} \chi_{287}(214,r) e\left(\frac{1 r + 2 r^{-1}}{287}\right) = -8.5976997208+4.9638842482i \)