Properties

Label 916.459
Modulus $916$
Conductor $4$
Order $2$
Real yes
Primitive no
Minimal yes
Parity odd

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(916, base_ring=CyclotomicField(2))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([1,0]))
 
pari: [g,chi] = znchar(Mod(459,916))
 

Basic properties

Modulus: \(916\)
Conductor: \(4\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(2\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: yes
Primitive: no, induced from \(\chi_{4}(3,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 916.b

\(\chi_{916}(459,\cdot)\)

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

Values on generators

\((459,693)\) → \((-1,1)\)

Values

\(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\(-1\)\(1\)\(-1\)\(1\)\(-1\)\(1\)\(-1\)\(1\)\(-1\)\(1\)\(-1\)\(1\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 916 }(459,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{916}(459,\cdot)) = \sum_{r\in \Z/916\Z} \chi_{916}(459,r) e\left(\frac{r}{458}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 916 }(459,·),\chi_{ 916 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{916}(459,\cdot),\chi_{916}(1,\cdot)) = \sum_{r\in \Z/916\Z} \chi_{916}(459,r) \chi_{916}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 916 }(459,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{916}(459,·)) = \sum_{r \in \Z/916\Z} \chi_{916}(459,r) e\left(\frac{1 r + 2 r^{-1}}{916}\right) = -3.9439791884i \)