Properties

Label 77.45
Modulus $77$
Conductor $7$
Order $6$
Real no
Primitive no
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, base_ring=CyclotomicField(6))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([1,0]))
 
pari: [g,chi] = znchar(Mod(45,77))
 

Basic properties

Modulus: \(77\)
Conductor: \(7\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(6\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{7}(3,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 77.g

\(\chi_{77}(12,\cdot)\) \(\chi_{77}(45,\cdot)\)

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

Related number fields

Field of values: \(\Q(\sqrt{-3}) \)
Fixed field: \(\Q(\zeta_{7})\)

Values on generators

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

Values

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

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 77 }(45,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{77}(45,\cdot)) = \sum_{r\in \Z/77\Z} \chi_{77}(45,r) e\left(\frac{2r}{77}\right) = -0.334452827+2.6245268729i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 77 }(45,·),\chi_{ 77 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{77}(45,\cdot),\chi_{77}(1,\cdot)) = \sum_{r\in \Z/77\Z} \chi_{77}(45,r) \chi_{77}(1,1-r) = -9 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 77 }(45,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{77}(45,·)) = \sum_{r \in \Z/77\Z} \chi_{77}(45,r) e\left(\frac{1 r + 2 r^{-1}}{77}\right) = 9.4095461105+-5.4326039798i \)