Properties

Label 403.315
Modulus $403$
Conductor $403$
Order $3$
Real no
Primitive yes
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(403, base_ring=CyclotomicField(6))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([2,4]))
 
pari: [g,chi] = znchar(Mod(315,403))
 

Basic properties

Modulus: \(403\)
Conductor: \(403\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(3\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 403.g

\(\chi_{403}(87,\cdot)\) \(\chi_{403}(315,\cdot)\)

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: 3.3.162409.2

Values on generators

\((249,313)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{2}{3}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{1}{3}\right)\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(1\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{2}{3}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 403 }(315,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{403}(315,\cdot)) = \sum_{r\in \Z/403\Z} \chi_{403}(315,r) e\left(\frac{2r}{403}\right) = 12.2528496935+15.9018135566i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 403 }(315,·),\chi_{ 403 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{403}(315,\cdot),\chi_{403}(1,\cdot)) = \sum_{r\in \Z/403\Z} \chi_{403}(315,r) \chi_{403}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 403 }(315,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{403}(315,·)) = \sum_{r \in \Z/403\Z} \chi_{403}(315,r) e\left(\frac{1 r + 2 r^{-1}}{403}\right) = 7.2812415105+12.6114802383i \)