Properties

Conductor 224
Order 24
Real no
Primitive yes
Minimal yes
Parity even
Orbit label 224.bd

Related objects

Learn more about

Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
 
sage: H = DirichletGroup_conrey(224)
 
sage: chi = H[37]
 
pari: [g,chi] = znchar(Mod(37,224))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 224
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 24
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = yes
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = even
Orbit label = 224.bd
Orbit index = 30

Galois orbit

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

\(\chi_{224}(37,\cdot)\) \(\chi_{224}(53,\cdot)\) \(\chi_{224}(93,\cdot)\) \(\chi_{224}(109,\cdot)\) \(\chi_{224}(149,\cdot)\) \(\chi_{224}(165,\cdot)\) \(\chi_{224}(205,\cdot)\) \(\chi_{224}(221,\cdot)\)

Values on generators

\((127,197,129)\) → \((1,e\left(\frac{1}{8}\right),e\left(\frac{1}{3}\right))\)

Values

-1135911131517192325
\(1\)\(1\)\(e\left(\frac{17}{24}\right)\)\(e\left(\frac{19}{24}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{23}{24}\right)\)\(e\left(\frac{7}{8}\right)\)\(-1\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{13}{24}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{7}{12}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{24})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 224 }(37,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{224}(37,\cdot)) = \sum_{r\in \Z/224\Z} \chi_{224}(37,r) e\left(\frac{r}{112}\right) = -0.0 \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 224 }(37,·),\chi_{ 224 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{224}(37,\cdot),\chi_{224}(1,\cdot)) = \sum_{r\in \Z/224\Z} \chi_{224}(37,r) \chi_{224}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 224 }(37,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{224}(37,·)) = \sum_{r \in \Z/224\Z} \chi_{224}(37,r) e\left(\frac{1 r + 2 r^{-1}}{224}\right) = 7.196248891+6.3109408992i \)