Properties

Conductor 389
Order 97
Real No
Primitive Yes
Parity Even
Orbit Label 389.d

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(389)
sage: chi = H[16]
pari: [g,chi] = znchar(Mod(16,389))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 389
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 97
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 = 389.d
Orbit index = 4

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 ]

\(\chi_{389}(5,\cdot)\) \(\chi_{389}(6,\cdot)\) \(\chi_{389}(7,\cdot)\) \(\chi_{389}(11,\cdot)\) \(\chi_{389}(13,\cdot)\) \(\chi_{389}(16,\cdot)\) \(\chi_{389}(17,\cdot)\) \(\chi_{389}(25,\cdot)\) \(\chi_{389}(30,\cdot)\) \(\chi_{389}(35,\cdot)\) \(\chi_{389}(36,\cdot)\) \(\chi_{389}(42,\cdot)\) \(\chi_{389}(49,\cdot)\) \(\chi_{389}(55,\cdot)\) \(\chi_{389}(58,\cdot)\) \(\chi_{389}(65,\cdot)\) \(\chi_{389}(66,\cdot)\) \(\chi_{389}(67,\cdot)\) \(\chi_{389}(69,\cdot)\) \(\chi_{389}(73,\cdot)\) \(\chi_{389}(74,\cdot)\) \(\chi_{389}(76,\cdot)\) \(\chi_{389}(77,\cdot)\) \(\chi_{389}(78,\cdot)\) \(\chi_{389}(79,\cdot)\) \(\chi_{389}(80,\cdot)\) \(\chi_{389}(81,\cdot)\) \(\chi_{389}(85,\cdot)\) \(\chi_{389}(91,\cdot)\) \(\chi_{389}(93,\cdot)\) ...

Values on generators

\(2\) → \(e\left(\frac{1}{97}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{1}{97}\right)\)\(e\left(\frac{77}{97}\right)\)\(e\left(\frac{2}{97}\right)\)\(e\left(\frac{84}{97}\right)\)\(e\left(\frac{78}{97}\right)\)\(e\left(\frac{37}{97}\right)\)\(e\left(\frac{3}{97}\right)\)\(e\left(\frac{57}{97}\right)\)\(e\left(\frac{85}{97}\right)\)\(e\left(\frac{75}{97}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{97})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 389 }(16,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{389}(16,\cdot)) = \sum_{r\in \Z/389\Z} \chi_{389}(16,r) e\left(\frac{2r}{389}\right) = 5.7653415612+-18.8616233841i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 389 }(16,·),\chi_{ 389 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{389}(16,\cdot),\chi_{389}(1,\cdot)) = \sum_{r\in \Z/389\Z} \chi_{389}(16,r) \chi_{389}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 389 }(16,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{389}(16,·)) = \sum_{r \in \Z/389\Z} \chi_{389}(16,r) e\left(\frac{1 r + 2 r^{-1}}{389}\right) = 5.2625434062+0.170500524i \)