Properties

Label 784.37
Modulus $784$
Conductor $784$
Order $84$
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(784, base_ring=CyclotomicField(84))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,21,64]))
 
pari: [g,chi] = znchar(Mod(37,784))
 

Basic properties

Modulus: \(784\)
Conductor: \(784\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
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 784.bt

\(\chi_{784}(37,\cdot)\) \(\chi_{784}(53,\cdot)\) \(\chi_{784}(93,\cdot)\) \(\chi_{784}(109,\cdot)\) \(\chi_{784}(149,\cdot)\) \(\chi_{784}(205,\cdot)\) \(\chi_{784}(221,\cdot)\) \(\chi_{784}(261,\cdot)\) \(\chi_{784}(277,\cdot)\) \(\chi_{784}(317,\cdot)\) \(\chi_{784}(333,\cdot)\) \(\chi_{784}(389,\cdot)\) \(\chi_{784}(429,\cdot)\) \(\chi_{784}(445,\cdot)\) \(\chi_{784}(485,\cdot)\) \(\chi_{784}(501,\cdot)\) \(\chi_{784}(541,\cdot)\) \(\chi_{784}(597,\cdot)\) \(\chi_{784}(613,\cdot)\) \(\chi_{784}(653,\cdot)\) \(\chi_{784}(669,\cdot)\) \(\chi_{784}(709,\cdot)\) \(\chi_{784}(725,\cdot)\) \(\chi_{784}(781,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Values on generators

\((687,197,689)\) → \((1,i,e\left(\frac{16}{21}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\(1\)\(1\)\(e\left(\frac{43}{84}\right)\)\(e\left(\frac{29}{84}\right)\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{61}{84}\right)\)\(e\left(\frac{25}{28}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{19}{42}\right)\)\(e\left(\frac{29}{42}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 784 }(37,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{784}(37,\cdot)) = \sum_{r\in \Z/784\Z} \chi_{784}(37,r) e\left(\frac{r}{392}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 784 }(37,·),\chi_{ 784 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{784}(37,\cdot),\chi_{784}(1,\cdot)) = \sum_{r\in \Z/784\Z} \chi_{784}(37,r) \chi_{784}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 784 }(37,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{784}(37,·)) = \sum_{r \in \Z/784\Z} \chi_{784}(37,r) e\left(\frac{1 r + 2 r^{-1}}{784}\right) = -8.464095805+-5.5417636576i \)