Properties

Conductor 287
Order 60
Real no
Primitive yes
Minimal yes
Parity even
Orbit label 287.bc

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[200]
 
pari: [g,chi] = znchar(Mod(200,287))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 287
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 60
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.bc
Orbit index = 29

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 ]
 

\(\chi_{287}(2,\cdot)\) \(\chi_{287}(39,\cdot)\) \(\chi_{287}(46,\cdot)\) \(\chi_{287}(74,\cdot)\) \(\chi_{287}(102,\cdot)\) \(\chi_{287}(121,\cdot)\) \(\chi_{287}(128,\cdot)\) \(\chi_{287}(144,\cdot)\) \(\chi_{287}(156,\cdot)\) \(\chi_{287}(172,\cdot)\) \(\chi_{287}(184,\cdot)\) \(\chi_{287}(200,\cdot)\) \(\chi_{287}(207,\cdot)\) \(\chi_{287}(226,\cdot)\) \(\chi_{287}(254,\cdot)\) \(\chi_{287}(282,\cdot)\)

Values on generators

\((206,211)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{1}{20}\right))\)

Values

-112345689101112
\(1\)\(1\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{49}{60}\right)\)\(e\left(\frac{41}{60}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{60})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 287 }(200,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{287}(200,\cdot)) = \sum_{r\in \Z/287\Z} \chi_{287}(200,r) e\left(\frac{2r}{287}\right) = -16.0068290977+5.5481007776i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 287 }(200,·),\chi_{ 287 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{287}(200,\cdot),\chi_{287}(1,\cdot)) = \sum_{r\in \Z/287\Z} \chi_{287}(200,r) \chi_{287}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 287 }(200,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{287}(200,·)) = \sum_{r \in \Z/287\Z} \chi_{287}(200,r) e\left(\frac{1 r + 2 r^{-1}}{287}\right) = 6.5555531623+-14.7240134759i \)