Properties

Conductor 77
Order 30
Real no
Primitive yes
Minimal yes
Parity even
Orbit label 77.n

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[17]
 
pari: [g,chi] = znchar(Mod(17,77))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 77
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 30
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = yes
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = even
Orbit label = 77.n
Orbit index = 14

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 ]
 

\(\chi_{77}(17,\cdot)\) \(\chi_{77}(19,\cdot)\) \(\chi_{77}(24,\cdot)\) \(\chi_{77}(40,\cdot)\) \(\chi_{77}(52,\cdot)\) \(\chi_{77}(61,\cdot)\) \(\chi_{77}(68,\cdot)\) \(\chi_{77}(73,\cdot)\)

Values on generators

\((45,57)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{9}{10}\right))\)

Values

-112345689101213
\(1\)\(1\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{2}{5}\right)\)
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)
 
\( \tau_{ a }( \chi_{ 77 }(17,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{77}(17,\cdot)) = \sum_{r\in \Z/77\Z} \chi_{77}(17,r) e\left(\frac{2r}{77}\right) = -6.1528512065+6.2563904954i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 77 }(17,·),\chi_{ 77 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{77}(17,\cdot),\chi_{77}(1,\cdot)) = \sum_{r\in \Z/77\Z} \chi_{77}(17,r) \chi_{77}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 77 }(17,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{77}(17,·)) = \sum_{r \in \Z/77\Z} \chi_{77}(17,r) e\left(\frac{1 r + 2 r^{-1}}{77}\right) = 15.1118771147+13.6067952711i \)