Properties

Conductor 35
Order 12
Real no
Primitive no
Minimal yes
Parity even
Orbit label 105.u

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(105)
 
sage: chi = H[52]
 
pari: [g,chi] = znchar(Mod(52,105))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 35
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 12
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 = 105.u
Orbit index = 21

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 ]
 

\(\chi_{105}(52,\cdot)\) \(\chi_{105}(73,\cdot)\) \(\chi_{105}(82,\cdot)\) \(\chi_{105}(103,\cdot)\)

Values on generators

\((71,22,31)\) → \((1,i,e\left(\frac{1}{6}\right))\)

Values

-1124811131617192223
\(1\)\(1\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{1}{6}\right)\)\(-i\)\(e\left(\frac{2}{3}\right)\)\(i\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{1}{3}\right)\)\(i\)\(e\left(\frac{1}{12}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{12})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 105 }(52,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{105}(52,\cdot)) = \sum_{r\in \Z/105\Z} \chi_{105}(52,r) e\left(\frac{2r}{105}\right) = -2.12208649+-5.5223861626i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 105 }(52,·),\chi_{ 105 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{105}(52,\cdot),\chi_{105}(1,\cdot)) = \sum_{r\in \Z/105\Z} \chi_{105}(52,r) \chi_{105}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 105 }(52,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{105}(52,·)) = \sum_{r \in \Z/105\Z} \chi_{105}(52,r) e\left(\frac{1 r + 2 r^{-1}}{105}\right) = 0.5988718849+-2.2350203018i \)