Properties

Label 693.487
Modulus $693$
Conductor $77$
Order $15$
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(693, base_ring=CyclotomicField(30))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,20,24]))
 
pari: [g,chi] = znchar(Mod(487,693))
 

Basic properties

Modulus: \(693\)
Conductor: \(77\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(15\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{77}(25,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 693.by

\(\chi_{693}(37,\cdot)\) \(\chi_{693}(163,\cdot)\) \(\chi_{693}(235,\cdot)\) \(\chi_{693}(289,\cdot)\) \(\chi_{693}(361,\cdot)\) \(\chi_{693}(478,\cdot)\) \(\chi_{693}(487,\cdot)\) \(\chi_{693}(676,\cdot)\)

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: 15.15.886528337182930278529.1

Values on generators

\((155,199,442)\) → \((1,e\left(\frac{2}{3}\right),e\left(\frac{4}{5}\right))\)

Values

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

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 693 }(487,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{693}(487,\cdot)) = \sum_{r\in \Z/693\Z} \chi_{693}(487,r) e\left(\frac{2r}{693}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 693 }(487,·),\chi_{ 693 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{693}(487,\cdot),\chi_{693}(1,\cdot)) = \sum_{r\in \Z/693\Z} \chi_{693}(487,r) \chi_{693}(1,1-r) = 3 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 693 }(487,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{693}(487,·)) = \sum_{r \in \Z/693\Z} \chi_{693}(487,r) e\left(\frac{1 r + 2 r^{-1}}{693}\right) = -0.0 \)