Properties

Label 77.3
Modulus $77$
Conductor $77$
Order $30$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(77)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([5,24]))
 
pari: [g,chi] = znchar(Mod(3,77))
 

Basic properties

Modulus: \(77\)
Conductor: \(77\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 77.p

\(\chi_{77}(3,\cdot)\) \(\chi_{77}(5,\cdot)\) \(\chi_{77}(26,\cdot)\) \(\chi_{77}(31,\cdot)\) \(\chi_{77}(38,\cdot)\) \(\chi_{77}(47,\cdot)\) \(\chi_{77}(59,\cdot)\) \(\chi_{77}(75,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Values on generators

\((45,57)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{4}{5}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(12\)\(13\)
\(-1\)\(1\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{3}{10}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{15})\)
Fixed field: 30.0.13209167403604364499542354001933559191813355687.1

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 77 }(3,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{77}(3,\cdot)) = \sum_{r\in \Z/77\Z} \chi_{77}(3,r) e\left(\frac{2r}{77}\right) = -8.5800168611+1.8393777925i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 77 }(3,·),\chi_{ 77 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{77}(3,\cdot),\chi_{77}(1,\cdot)) = \sum_{r\in \Z/77\Z} \chi_{77}(3,r) \chi_{77}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 77 }(3,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{77}(3,·)) = \sum_{r \in \Z/77\Z} \chi_{77}(3,r) e\left(\frac{1 r + 2 r^{-1}}{77}\right) = -0.6856395767+1.5399717028i \)