Properties

Conductor 77
Order 15
Real No
Primitive Yes
Parity Even
Orbit Label 77.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(77)
sage: chi = H[16]
pari: [g,chi] = znchar(Mod(16,77))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 77
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 15
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 = 77.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 ]

\(\chi_{77}(4,\cdot)\) \(\chi_{77}(9,\cdot)\) \(\chi_{77}(16,\cdot)\) \(\chi_{77}(25,\cdot)\) \(\chi_{77}(37,\cdot)\) \(\chi_{77}(53,\cdot)\) \(\chi_{77}(58,\cdot)\) \(\chi_{77}(60,\cdot)\)

Values on generators

\((45,57)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{2}{5}\right))\)

Values

-112345689101213
\(1\)\(1\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{2}{5}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{15})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 77 }(16,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{77}(16,\cdot)) = \sum_{r\in \Z/77\Z} \chi_{77}(16,r) e\left(\frac{2r}{77}\right) = 5.7558197842+6.6234838727i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 77 }(16,·),\chi_{ 77 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{77}(16,\cdot),\chi_{77}(1,\cdot)) = \sum_{r\in \Z/77\Z} \chi_{77}(16,r) \chi_{77}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 77 }(16,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{77}(16,·)) = \sum_{r \in \Z/77\Z} \chi_{77}(16,r) e\left(\frac{1 r + 2 r^{-1}}{77}\right) = 4.9221449043+1.0462341969i \)