Properties

Label 144.19
Modulus $144$
Conductor $16$
Order $4$
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(144)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([2,3,0]))
 
pari: [g,chi] = znchar(Mod(19,144))
 

Basic properties

Modulus: \(144\)
Conductor: \(16\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(4\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{16}(3,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 144.m

\(\chi_{144}(19,\cdot)\) \(\chi_{144}(91,\cdot)\)

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

\((127,37,65)\) → \((-1,-i,1)\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(-1\)\(1\)\(-i\)\(1\)\(i\)\(i\)\(1\)\(-i\)\(1\)\(-1\)\(i\)\(-1\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\sqrt{-1}) \)
Fixed field: 4.0.2048.2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 144 }(19,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{144}(19,\cdot)) = \sum_{r\in \Z/144\Z} \chi_{144}(19,r) e\left(\frac{r}{72}\right) = -0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 144 }(19,·),\chi_{ 144 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{144}(19,\cdot),\chi_{144}(1,\cdot)) = \sum_{r\in \Z/144\Z} \chi_{144}(19,r) \chi_{144}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 144 }(19,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{144}(19,·)) = \sum_{r \in \Z/144\Z} \chi_{144}(19,r) e\left(\frac{1 r + 2 r^{-1}}{144}\right) = -0.0 \)