Properties

Label 416.179
Modulus $416$
Conductor $416$
Order $24$
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(416, base_ring=CyclotomicField(24))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([12,21,20]))
 
pari: [g,chi] = znchar(Mod(179,416))
 

Basic properties

Modulus: \(416\)
Conductor: \(416\)
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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 416.cb

\(\chi_{416}(43,\cdot)\) \(\chi_{416}(75,\cdot)\) \(\chi_{416}(147,\cdot)\) \(\chi_{416}(179,\cdot)\) \(\chi_{416}(251,\cdot)\) \(\chi_{416}(283,\cdot)\) \(\chi_{416}(355,\cdot)\) \(\chi_{416}(387,\cdot)\)

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

\((287,261,353)\) → \((-1,e\left(\frac{7}{8}\right),e\left(\frac{5}{6}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(15\)\(17\)\(19\)\(21\)\(23\)
\(-1\)\(1\)\(e\left(\frac{11}{24}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{17}{24}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{19}{24}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{1}{12}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{24})\)
Fixed field: 24.0.188216044816745326913150945765287080795084676923392.1

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 416 }(179,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{416}(179,\cdot)) = \sum_{r\in \Z/416\Z} \chi_{416}(179,r) e\left(\frac{r}{208}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 416 }(179,·),\chi_{ 416 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{416}(179,\cdot),\chi_{416}(1,\cdot)) = \sum_{r\in \Z/416\Z} \chi_{416}(179,r) \chi_{416}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 416 }(179,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{416}(179,·)) = \sum_{r \in \Z/416\Z} \chi_{416}(179,r) e\left(\frac{1 r + 2 r^{-1}}{416}\right) = 21.3427286066+1.3988763388i \)