Properties

Conductor 287
Order 120
Real No
Primitive Yes
Parity Even
Orbit Label 287.be

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[12]
 
pari: [g,chi] = znchar(Mod(12,287))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 287
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 120
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.be
Orbit index = 31

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 ]
 

\(\chi_{287}(12,\cdot)\) \(\chi_{287}(17,\cdot)\) \(\chi_{287}(19,\cdot)\) \(\chi_{287}(24,\cdot)\) \(\chi_{287}(26,\cdot)\) \(\chi_{287}(47,\cdot)\) \(\chi_{287}(52,\cdot)\) \(\chi_{287}(54,\cdot)\) \(\chi_{287}(75,\cdot)\) \(\chi_{287}(89,\cdot)\) \(\chi_{287}(94,\cdot)\) \(\chi_{287}(101,\cdot)\) \(\chi_{287}(108,\cdot)\) \(\chi_{287}(110,\cdot)\) \(\chi_{287}(117,\cdot)\) \(\chi_{287}(129,\cdot)\) \(\chi_{287}(136,\cdot)\) \(\chi_{287}(138,\cdot)\) \(\chi_{287}(145,\cdot)\) \(\chi_{287}(152,\cdot)\) \(\chi_{287}(157,\cdot)\) \(\chi_{287}(171,\cdot)\) \(\chi_{287}(192,\cdot)\) \(\chi_{287}(194,\cdot)\) \(\chi_{287}(199,\cdot)\) \(\chi_{287}(220,\cdot)\) \(\chi_{287}(222,\cdot)\) \(\chi_{287}(227,\cdot)\) \(\chi_{287}(229,\cdot)\) \(\chi_{287}(234,\cdot)\) ...

Values on generators

\((206,211)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{27}{40}\right))\)

Values

-112345689101112
\(1\)\(1\)\(e\left(\frac{13}{60}\right)\)\(e\left(\frac{23}{24}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{1}{60}\right)\)\(e\left(\frac{7}{40}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{43}{120}\right)\)\(e\left(\frac{47}{120}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{120})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 287 }(12,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{287}(12,\cdot)) = \sum_{r\in \Z/287\Z} \chi_{287}(12,r) e\left(\frac{2r}{287}\right) = 16.9049434389+1.1058423614i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 287 }(12,·),\chi_{ 287 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{287}(12,\cdot),\chi_{287}(1,\cdot)) = \sum_{r\in \Z/287\Z} \chi_{287}(12,r) \chi_{287}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 287 }(12,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{287}(12,·)) = \sum_{r \in \Z/287\Z} \chi_{287}(12,r) e\left(\frac{1 r + 2 r^{-1}}{287}\right) = 4.6914979695+3.7991001475i \)