Properties

Label 288.13
Modulus $288$
Conductor $288$
Order $24$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(288, base_ring=CyclotomicField(24))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,21,8]))
 
pari: [g,chi] = znchar(Mod(13,288))
 

Basic properties

Modulus: \(288\)
Conductor: \(288\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(24\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 288.bc

\(\chi_{288}(13,\cdot)\) \(\chi_{288}(61,\cdot)\) \(\chi_{288}(85,\cdot)\) \(\chi_{288}(133,\cdot)\) \(\chi_{288}(157,\cdot)\) \(\chi_{288}(205,\cdot)\) \(\chi_{288}(229,\cdot)\) \(\chi_{288}(277,\cdot)\)

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: \(\Q(\zeta_{24})\)
Fixed field: 24.24.18351423083070806589199715754737431920771072.1

Values on generators

\((127,37,65)\) → \((1,e\left(\frac{7}{8}\right),e\left(\frac{1}{3}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(1\)\(1\)\(e\left(\frac{13}{24}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{17}{24}\right)\)\(e\left(\frac{19}{24}\right)\)\(-1\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{23}{24}\right)\)\(e\left(\frac{2}{3}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 288 }(13,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{288}(13,\cdot)) = \sum_{r\in \Z/288\Z} \chi_{288}(13,r) e\left(\frac{r}{144}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 288 }(13,·),\chi_{ 288 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{288}(13,\cdot),\chi_{288}(1,\cdot)) = \sum_{r\in \Z/288\Z} \chi_{288}(13,r) \chi_{288}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 288 }(13,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{288}(13,·)) = \sum_{r \in \Z/288\Z} \chi_{288}(13,r) e\left(\frac{1 r + 2 r^{-1}}{288}\right) = 12.7591446296+-11.1894695281i \)