Properties

Label 140.9
Modulus $140$
Conductor $35$
Order $6$
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(140, base_ring=CyclotomicField(6))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,3,2]))
 
pari: [g,chi] = znchar(Mod(9,140))
 

Basic properties

Modulus: \(140\)
Conductor: \(35\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(6\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{35}(9,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 140.q

\(\chi_{140}(9,\cdot)\) \(\chi_{140}(109,\cdot)\)

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: 6.6.300125.1

Values on generators

\((71,57,101)\) → \((1,-1,e\left(\frac{1}{3}\right))\)

Values

\(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\(1\)\(1\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(-1\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{6}\right)\)\(-1\)\(1\)\(e\left(\frac{1}{3}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 140 }(9,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{140}(9,\cdot)) = \sum_{r\in \Z/140\Z} \chi_{140}(9,r) e\left(\frac{r}{70}\right) = 0.749363631+-11.8084060799i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 140 }(9,·),\chi_{ 140 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{140}(9,\cdot),\chi_{140}(1,\cdot)) = \sum_{r\in \Z/140\Z} \chi_{140}(9,r) \chi_{140}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 140 }(9,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{140}(9,·)) = \sum_{r \in \Z/140\Z} \chi_{140}(9,r) e\left(\frac{1 r + 2 r^{-1}}{140}\right) = 0.0 \)