Properties

Conductor 73
Order 72
Real No
Primitive Yes
Parity Odd
Orbit Label 73.l

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(73)
sage: chi = H[28]
pari: [g,chi] = znchar(Mod(28,73))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 73
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 72
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 = 73.l
Orbit index = 12

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 ]

\(\chi_{73}(5,\cdot)\) \(\chi_{73}(11,\cdot)\) \(\chi_{73}(13,\cdot)\) \(\chi_{73}(14,\cdot)\) \(\chi_{73}(15,\cdot)\) \(\chi_{73}(20,\cdot)\) \(\chi_{73}(26,\cdot)\) \(\chi_{73}(28,\cdot)\) \(\chi_{73}(29,\cdot)\) \(\chi_{73}(31,\cdot)\) \(\chi_{73}(33,\cdot)\) \(\chi_{73}(34,\cdot)\) \(\chi_{73}(39,\cdot)\) \(\chi_{73}(40,\cdot)\) \(\chi_{73}(42,\cdot)\) \(\chi_{73}(44,\cdot)\) \(\chi_{73}(45,\cdot)\) \(\chi_{73}(47,\cdot)\) \(\chi_{73}(53,\cdot)\) \(\chi_{73}(58,\cdot)\) \(\chi_{73}(59,\cdot)\) \(\chi_{73}(60,\cdot)\) \(\chi_{73}(62,\cdot)\) \(\chi_{73}(68,\cdot)\)

Values on generators

\(5\) → \(e\left(\frac{49}{72}\right)\)

Values

-11234567891011
\(-1\)\(1\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{49}{72}\right)\)\(e\left(\frac{19}{36}\right)\)\(e\left(\frac{11}{24}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{31}{72}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{72})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 73 }(28,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{73}(28,\cdot)) = \sum_{r\in \Z/73\Z} \chi_{73}(28,r) e\left(\frac{2r}{73}\right) = 2.9412107022+8.0218002721i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 73 }(28,·),\chi_{ 73 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{73}(28,\cdot),\chi_{73}(1,\cdot)) = \sum_{r\in \Z/73\Z} \chi_{73}(28,r) \chi_{73}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 73 }(28,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{73}(28,·)) = \sum_{r \in \Z/73\Z} \chi_{73}(28,r) e\left(\frac{1 r + 2 r^{-1}}{73}\right) = 14.303505431+-2.5220939262i \)