Properties

Label 693.445
Modulus $693$
Conductor $693$
Order $15$
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(693, base_ring=CyclotomicField(30))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([10,20,12]))
 
pari: [g,chi] = znchar(Mod(445,693))
 

Basic properties

Modulus: \(693\)
Conductor: \(693\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(15\)
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 693.bx

\(\chi_{693}(4,\cdot)\) \(\chi_{693}(16,\cdot)\) \(\chi_{693}(130,\cdot)\) \(\chi_{693}(256,\cdot)\) \(\chi_{693}(268,\cdot)\) \(\chi_{693}(394,\cdot)\) \(\chi_{693}(445,\cdot)\) \(\chi_{693}(520,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: 15.15.3091133177133909578645502426129.1

Values on generators

\((155,199,442)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{2}{3}\right),e\left(\frac{2}{5}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(13\)\(16\)\(17\)\(19\)\(20\)
\(1\)\(1\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{11}{15}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 693 }(445,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{693}(445,\cdot)) = \sum_{r\in \Z/693\Z} \chi_{693}(445,r) e\left(\frac{2r}{693}\right) = 18.3702349987+18.8556216045i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 693 }(445,·),\chi_{ 693 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{693}(445,\cdot),\chi_{693}(1,\cdot)) = \sum_{r\in \Z/693\Z} \chi_{693}(445,r) \chi_{693}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 693 }(445,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{693}(445,·)) = \sum_{r \in \Z/693\Z} \chi_{693}(445,r) e\left(\frac{1 r + 2 r^{-1}}{693}\right) = -46.1571031501+-9.8109951422i \)