Properties

Label 105.46
Modulus $105$
Conductor $7$
Order $3$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(105, base_ring=CyclotomicField(6))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,0,4]))
 
pari: [g,chi] = znchar(Mod(46,105))
 

Basic properties

Modulus: \(105\)
Conductor: \(7\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(3\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{7}(4,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 105.i

\(\chi_{105}(16,\cdot)\) \(\chi_{105}(46,\cdot)\)

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

Related number fields

Field of values: \(\Q(\sqrt{-3}) \)
Fixed field: \(\Q(\zeta_{7})^+\)

Values on generators

\((71,22,31)\) → \((1,1,e\left(\frac{2}{3}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(8\)\(11\)\(13\)\(16\)\(17\)\(19\)\(22\)\(23\)
\(1\)\(1\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(1\)\(e\left(\frac{2}{3}\right)\)\(1\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(1\)\(e\left(\frac{1}{3}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 105 }(46,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{105}(46,\cdot)) = \sum_{r\in \Z/105\Z} \chi_{105}(46,r) e\left(\frac{2r}{105}\right) = -0.1675628019+-2.6404398701i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 105 }(46,·),\chi_{ 105 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{105}(46,\cdot),\chi_{105}(1,\cdot)) = \sum_{r\in \Z/105\Z} \chi_{105}(46,r) \chi_{105}(1,1-r) = -3 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 105 }(46,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{105}(46,·)) = \sum_{r \in \Z/105\Z} \chi_{105}(46,r) e\left(\frac{1 r + 2 r^{-1}}{105}\right) = 9.8665037685+17.0892858201i \)