Properties

Conductor 77
Order 30
Real No
Primitive Yes
Parity Odd
Orbit Label 77.o

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[74]
 
pari: [g,chi] = znchar(Mod(74,77))
 

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
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = Odd
Orbit label = 77.o
Orbit index = 15

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 ]
 

\(\chi_{77}(2,\cdot)\) \(\chi_{77}(18,\cdot)\) \(\chi_{77}(30,\cdot)\) \(\chi_{77}(39,\cdot)\) \(\chi_{77}(46,\cdot)\) \(\chi_{77}(51,\cdot)\) \(\chi_{77}(72,\cdot)\) \(\chi_{77}(74,\cdot)\)

Values on generators

\((45,57)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{3}{10}\right))\)

Values

-112345689101213
\(-1\)\(1\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{3}{10}\right)\)
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)
 
\( \tau_{ a }( \chi_{ 77 }(74,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{77}(74,\cdot)) = \sum_{r\in \Z/77\Z} \chi_{77}(74,r) e\left(\frac{2r}{77}\right) = 3.6183988583+-7.9941972519i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 77 }(74,·),\chi_{ 77 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{77}(74,\cdot),\chi_{77}(1,\cdot)) = \sum_{r\in \Z/77\Z} \chi_{77}(74,r) \chi_{77}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 77 }(74,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{77}(74,·)) = \sum_{r \in \Z/77\Z} \chi_{77}(74,r) e\left(\frac{1 r + 2 r^{-1}}{77}\right) = 2.5146560404+1.1195970029i \)