Properties

Conductor 19
Order 18
Real No
Primitive No
Parity Odd
Orbit Label 152.r

Related objects

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Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
 
sage: H = DirichletGroup_conrey(152)
 
sage: chi = H[33]
 
pari: [g,chi] = znchar(Mod(33,152))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 19
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 18
Real = No
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = No
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = Odd
Orbit label = 152.r
Orbit index = 18

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 ]
 

\(\chi_{152}(33,\cdot)\) \(\chi_{152}(41,\cdot)\) \(\chi_{152}(89,\cdot)\) \(\chi_{152}(97,\cdot)\) \(\chi_{152}(105,\cdot)\) \(\chi_{152}(129,\cdot)\)

Inducing primitive character

\(\chi_{19}(14,\cdot)\)

Values on generators

\((39,77,97)\) → \((1,1,e\left(\frac{7}{18}\right))\)

Values

-113579111315172123
\(-1\)\(1\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{7}{9}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{9})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 152 }(33,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{152}(33,\cdot)) = \sum_{r\in \Z/152\Z} \chi_{152}(33,r) e\left(\frac{r}{76}\right) = 0.0 \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 152 }(33,·),\chi_{ 152 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{152}(33,\cdot),\chi_{152}(1,\cdot)) = \sum_{r\in \Z/152\Z} \chi_{152}(33,r) \chi_{152}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 152 }(33,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{152}(33,·)) = \sum_{r \in \Z/152\Z} \chi_{152}(33,r) e\left(\frac{1 r + 2 r^{-1}}{152}\right) = 0.0 \)