Properties

Label 76.13
Modulus $76$
Conductor $19$
Order $18$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(76, base_ring=CyclotomicField(18))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,5]))
 
pari: [g,chi] = znchar(Mod(13,76))
 

Basic properties

Modulus: \(76\)
Conductor: \(19\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(18\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{19}(13,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 76.j

\(\chi_{76}(13,\cdot)\) \(\chi_{76}(21,\cdot)\) \(\chi_{76}(29,\cdot)\) \(\chi_{76}(33,\cdot)\) \(\chi_{76}(41,\cdot)\) \(\chi_{76}(53,\cdot)\)

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: \(\Q(\zeta_{9})\)
Fixed field: \(\Q(\zeta_{19})\)

Values on generators

\((39,21)\) → \((1,e\left(\frac{5}{18}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(21\)\(23\)
\(-1\)\(1\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{5}{9}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 76 }(13,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{76}(13,\cdot)) = \sum_{r\in \Z/76\Z} \chi_{76}(13,r) e\left(\frac{r}{38}\right) = -5.6080501684+-6.6745616567i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 76 }(13,·),\chi_{ 76 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{76}(13,\cdot),\chi_{76}(1,\cdot)) = \sum_{r\in \Z/76\Z} \chi_{76}(13,r) \chi_{76}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 76 }(13,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{76}(13,·)) = \sum_{r \in \Z/76\Z} \chi_{76}(13,r) e\left(\frac{1 r + 2 r^{-1}}{76}\right) = -0.0 \)