Properties

Conductor 53
Order 13
Real No
Primitive Yes
Parity Even
Orbit Label 53.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(53)
sage: chi = H[15]
pari: [g,chi] = znchar(Mod(15,53))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 53
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 13
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 = 53.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 ]

\(\chi_{53}(10,\cdot)\) \(\chi_{53}(13,\cdot)\) \(\chi_{53}(15,\cdot)\) \(\chi_{53}(16,\cdot)\) \(\chi_{53}(24,\cdot)\) \(\chi_{53}(28,\cdot)\) \(\chi_{53}(36,\cdot)\) \(\chi_{53}(42,\cdot)\) \(\chi_{53}(44,\cdot)\) \(\chi_{53}(46,\cdot)\) \(\chi_{53}(47,\cdot)\) \(\chi_{53}(49,\cdot)\)

Values on generators

\(2\) → \(e\left(\frac{3}{13}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{3}{13}\right)\)\(e\left(\frac{12}{13}\right)\)\(e\left(\frac{6}{13}\right)\)\(e\left(\frac{11}{13}\right)\)\(e\left(\frac{2}{13}\right)\)\(e\left(\frac{3}{13}\right)\)\(e\left(\frac{9}{13}\right)\)\(e\left(\frac{11}{13}\right)\)\(e\left(\frac{1}{13}\right)\)\(e\left(\frac{5}{13}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{13})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 53 }(15,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{53}(15,\cdot)) = \sum_{r\in \Z/53\Z} \chi_{53}(15,r) e\left(\frac{2r}{53}\right) = 5.5573391939+-4.7027631328i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 53 }(15,·),\chi_{ 53 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{53}(15,\cdot),\chi_{53}(1,\cdot)) = \sum_{r\in \Z/53\Z} \chi_{53}(15,r) \chi_{53}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 53 }(15,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{53}(15,·)) = \sum_{r \in \Z/53\Z} \chi_{53}(15,r) e\left(\frac{1 r + 2 r^{-1}}{53}\right) = -9.4543013852+-8.3757801497i \)