Properties

Conductor 463
Order 77
Real No
Primitive Yes
Parity Even
Orbit Label 463.m

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(463)
sage: chi = H[58]
pari: [g,chi] = znchar(Mod(58,463))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 463
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 77
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 = 463.m
Orbit index = 13

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 ]

\(\chi_{463}(8,\cdot)\) \(\chi_{463}(18,\cdot)\) \(\chi_{463}(47,\cdot)\) \(\chi_{463}(49,\cdot)\) \(\chi_{463}(57,\cdot)\) \(\chi_{463}(58,\cdot)\) \(\chi_{463}(64,\cdot)\) \(\chi_{463}(65,\cdot)\) \(\chi_{463}(66,\cdot)\) \(\chi_{463}(70,\cdot)\) \(\chi_{463}(78,\cdot)\) \(\chi_{463}(84,\cdot)\) \(\chi_{463}(86,\cdot)\) \(\chi_{463}(97,\cdot)\) \(\chi_{463}(100,\cdot)\) \(\chi_{463}(111,\cdot)\) \(\chi_{463}(120,\cdot)\) \(\chi_{463}(123,\cdot)\) \(\chi_{463}(124,\cdot)\) \(\chi_{463}(144,\cdot)\) \(\chi_{463}(146,\cdot)\) \(\chi_{463}(149,\cdot)\) \(\chi_{463}(159,\cdot)\) \(\chi_{463}(161,\cdot)\) \(\chi_{463}(189,\cdot)\) \(\chi_{463}(209,\cdot)\) \(\chi_{463}(226,\cdot)\) \(\chi_{463}(242,\cdot)\) \(\chi_{463}(244,\cdot)\) \(\chi_{463}(262,\cdot)\) ...

Values on generators

\(3\) → \(e\left(\frac{60}{77}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{38}{77}\right)\)\(e\left(\frac{60}{77}\right)\)\(e\left(\frac{76}{77}\right)\)\(e\left(\frac{31}{77}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{17}{77}\right)\)\(e\left(\frac{37}{77}\right)\)\(e\left(\frac{43}{77}\right)\)\(e\left(\frac{69}{77}\right)\)\(e\left(\frac{39}{77}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{77})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 463 }(58,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{463}(58,\cdot)) = \sum_{r\in \Z/463\Z} \chi_{463}(58,r) e\left(\frac{2r}{463}\right) = -6.4890106934+-20.5156706013i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 463 }(58,·),\chi_{ 463 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{463}(58,\cdot),\chi_{463}(1,\cdot)) = \sum_{r\in \Z/463\Z} \chi_{463}(58,r) \chi_{463}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 463 }(58,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{463}(58,·)) = \sum_{r \in \Z/463\Z} \chi_{463}(58,r) e\left(\frac{1 r + 2 r^{-1}}{463}\right) = 0.0252813372+1.239112216i \)