Properties

Label 240.227
Modulus $240$
Conductor $240$
Order $4$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(240, base_ring=CyclotomicField(4))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([2,3,2,1]))
 
pari: [g,chi] = znchar(Mod(227,240))
 

Basic properties

Modulus: \(240\)
Conductor: \(240\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(4\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 240.bd

\(\chi_{240}(203,\cdot)\) \(\chi_{240}(227,\cdot)\)

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: \(\Q(\sqrt{-1}) \)
Fixed field: 4.0.2304000.2

Values on generators

\((31,181,161,97)\) → \((-1,-i,-1,i)\)

Values

\(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\(-1\)\(1\)\(i\)\(-i\)\(1\)\(-i\)\(i\)\(i\)\(i\)\(-1\)\(1\)\(1\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 240 }(227,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{240}(227,\cdot)) = \sum_{r\in \Z/240\Z} \chi_{240}(227,r) e\left(\frac{r}{120}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 240 }(227,·),\chi_{ 240 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{240}(227,\cdot),\chi_{240}(1,\cdot)) = \sum_{r\in \Z/240\Z} \chi_{240}(227,r) \chi_{240}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 240 }(227,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{240}(227,·)) = \sum_{r \in \Z/240\Z} \chi_{240}(227,r) e\left(\frac{1 r + 2 r^{-1}}{240}\right) = -0.0 \)