Properties

Conductor 367
Order 61
Real No
Primitive Yes
Parity Even
Orbit Label 367.e

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(367)
sage: chi = H[49]
pari: [g,chi] = znchar(Mod(49,367))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 367
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 61
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 = 367.e
Orbit index = 5

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 ]

\(\chi_{367}(7,\cdot)\) \(\chi_{367}(8,\cdot)\) \(\chi_{367}(9,\cdot)\) \(\chi_{367}(15,\cdot)\) \(\chi_{367}(25,\cdot)\) \(\chi_{367}(46,\cdot)\) \(\chi_{367}(47,\cdot)\) \(\chi_{367}(49,\cdot)\) \(\chi_{367}(52,\cdot)\) \(\chi_{367}(56,\cdot)\) \(\chi_{367}(59,\cdot)\) \(\chi_{367}(63,\cdot)\) \(\chi_{367}(64,\cdot)\) \(\chi_{367}(72,\cdot)\) \(\chi_{367}(74,\cdot)\) \(\chi_{367}(81,\cdot)\) \(\chi_{367}(87,\cdot)\) \(\chi_{367}(101,\cdot)\) \(\chi_{367}(105,\cdot)\) \(\chi_{367}(106,\cdot)\) \(\chi_{367}(107,\cdot)\) \(\chi_{367}(114,\cdot)\) \(\chi_{367}(120,\cdot)\) \(\chi_{367}(122,\cdot)\) \(\chi_{367}(124,\cdot)\) \(\chi_{367}(132,\cdot)\) \(\chi_{367}(134,\cdot)\) \(\chi_{367}(135,\cdot)\) \(\chi_{367}(137,\cdot)\) \(\chi_{367}(145,\cdot)\) ...

Values on generators

\(6\) → \(e\left(\frac{4}{61}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{9}{61}\right)\)\(e\left(\frac{56}{61}\right)\)\(e\left(\frac{18}{61}\right)\)\(e\left(\frac{31}{61}\right)\)\(e\left(\frac{4}{61}\right)\)\(e\left(\frac{48}{61}\right)\)\(e\left(\frac{27}{61}\right)\)\(e\left(\frac{51}{61}\right)\)\(e\left(\frac{40}{61}\right)\)\(e\left(\frac{42}{61}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{61})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 367 }(49,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{367}(49,\cdot)) = \sum_{r\in \Z/367\Z} \chi_{367}(49,r) e\left(\frac{2r}{367}\right) = 7.7898810406+-17.5019357036i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 367 }(49,·),\chi_{ 367 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{367}(49,\cdot),\chi_{367}(1,\cdot)) = \sum_{r\in \Z/367\Z} \chi_{367}(49,r) \chi_{367}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 367 }(49,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{367}(49,·)) = \sum_{r \in \Z/367\Z} \chi_{367}(49,r) e\left(\frac{1 r + 2 r^{-1}}{367}\right) = -21.8723738096+-10.9325252006i \)