Properties

Conductor 215
Order 84
Real no
Primitive no
Minimal yes
Parity even
Orbit label 430.x

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(430)
 
sage: chi = H[3]
 
pari: [g,chi] = znchar(Mod(3,430))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 215
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 84
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = no
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = even
Orbit label = 430.x
Orbit index = 24

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 ]
 

\(\chi_{430}(3,\cdot)\) \(\chi_{430}(33,\cdot)\) \(\chi_{430}(63,\cdot)\) \(\chi_{430}(73,\cdot)\) \(\chi_{430}(77,\cdot)\) \(\chi_{430}(147,\cdot)\) \(\chi_{430}(157,\cdot)\) \(\chi_{430}(163,\cdot)\) \(\chi_{430}(177,\cdot)\) \(\chi_{430}(227,\cdot)\) \(\chi_{430}(233,\cdot)\) \(\chi_{430}(243,\cdot)\) \(\chi_{430}(263,\cdot)\) \(\chi_{430}(277,\cdot)\) \(\chi_{430}(287,\cdot)\) \(\chi_{430}(313,\cdot)\) \(\chi_{430}(327,\cdot)\) \(\chi_{430}(347,\cdot)\) \(\chi_{430}(363,\cdot)\) \(\chi_{430}(373,\cdot)\) \(\chi_{430}(377,\cdot)\) \(\chi_{430}(407,\cdot)\) \(\chi_{430}(413,\cdot)\) \(\chi_{430}(417,\cdot)\)

Values on generators

\((87,261)\) → \((-i,e\left(\frac{1}{42}\right))\)

Values

-1137911131719212327
\(1\)\(1\)\(e\left(\frac{23}{84}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{23}{42}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{1}{84}\right)\)\(e\left(\frac{55}{84}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{53}{84}\right)\)\(e\left(\frac{23}{28}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{84})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 430 }(3,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{430}(3,\cdot)) = \sum_{r\in \Z/430\Z} \chi_{430}(3,r) e\left(\frac{r}{215}\right) = 14.3041602633+-3.2235072764i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 430 }(3,·),\chi_{ 430 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{430}(3,\cdot),\chi_{430}(1,\cdot)) = \sum_{r\in \Z/430\Z} \chi_{430}(3,r) \chi_{430}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 430 }(3,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{430}(3,·)) = \sum_{r \in \Z/430\Z} \chi_{430}(3,r) e\left(\frac{1 r + 2 r^{-1}}{430}\right) = -0.9272289619+-2.6498681308i \)