Properties

Conductor 311
Order 62
Real No
Primitive Yes
Parity Odd
Orbit Label 311.f

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(311)
sage: chi = H[61]
pari: [g,chi] = znchar(Mod(61,311))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 311
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 62
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 = Odd
Orbit label = 311.f
Orbit index = 6

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 ]

\(\chi_{311}(11,\cdot)\) \(\chi_{311}(41,\cdot)\) \(\chi_{311}(46,\cdot)\) \(\chi_{311}(51,\cdot)\) \(\chi_{311}(61,\cdot)\) \(\chi_{311}(68,\cdot)\) \(\chi_{311}(77,\cdot)\) \(\chi_{311}(86,\cdot)\) \(\chi_{311}(87,\cdot)\) \(\chi_{311}(116,\cdot)\) \(\chi_{311}(142,\cdot)\) \(\chi_{311}(143,\cdot)\) \(\chi_{311}(165,\cdot)\) \(\chi_{311}(171,\cdot)\) \(\chi_{311}(185,\cdot)\) \(\chi_{311}(190,\cdot)\) \(\chi_{311}(198,\cdot)\) \(\chi_{311}(206,\cdot)\) \(\chi_{311}(220,\cdot)\) \(\chi_{311}(222,\cdot)\) \(\chi_{311}(228,\cdot)\) \(\chi_{311}(262,\cdot)\) \(\chi_{311}(264,\cdot)\) \(\chi_{311}(279,\cdot)\) \(\chi_{311}(287,\cdot)\) \(\chi_{311}(291,\cdot)\) \(\chi_{311}(293,\cdot)\) \(\chi_{311}(296,\cdot)\) \(\chi_{311}(298,\cdot)\) \(\chi_{311}(304,\cdot)\)

Values on generators

\(17\) → \(e\left(\frac{29}{62}\right)\)

Values

-11234567891011
\(-1\)\(1\)\(e\left(\frac{9}{31}\right)\)\(e\left(\frac{22}{31}\right)\)\(e\left(\frac{18}{31}\right)\)\(e\left(\frac{21}{31}\right)\)\(1\)\(e\left(\frac{15}{31}\right)\)\(e\left(\frac{27}{31}\right)\)\(e\left(\frac{13}{31}\right)\)\(e\left(\frac{30}{31}\right)\)\(e\left(\frac{9}{62}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{31})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 311 }(61,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{311}(61,\cdot)) = \sum_{r\in \Z/311\Z} \chi_{311}(61,r) e\left(\frac{2r}{311}\right) = 17.1396807377+4.1510654307i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 311 }(61,·),\chi_{ 311 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{311}(61,\cdot),\chi_{311}(1,\cdot)) = \sum_{r\in \Z/311\Z} \chi_{311}(61,r) \chi_{311}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 311 }(61,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{311}(61,·)) = \sum_{r \in \Z/311\Z} \chi_{311}(61,r) e\left(\frac{1 r + 2 r^{-1}}{311}\right) = 12.5509276803+-9.7151411719i \)