Properties

Label 647.53
Modulus $647$
Conductor $647$
Order $17$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(647, base_ring=CyclotomicField(34))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([24]))
 
pari: [g,chi] = znchar(Mod(53,647))
 

Basic properties

Modulus: \(647\)
Conductor: \(647\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(17\)
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 647.c

\(\chi_{647}(43,\cdot)\) \(\chi_{647}(53,\cdot)\) \(\chi_{647}(67,\cdot)\) \(\chi_{647}(218,\cdot)\) \(\chi_{647}(221,\cdot)\) \(\chi_{647}(293,\cdot)\) \(\chi_{647}(300,\cdot)\) \(\chi_{647}(306,\cdot)\) \(\chi_{647}(316,\cdot)\) \(\chi_{647}(338,\cdot)\) \(\chi_{647}(372,\cdot)\) \(\chi_{647}(445,\cdot)\) \(\chi_{647}(468,\cdot)\) \(\chi_{647}(555,\cdot)\) \(\chi_{647}(573,\cdot)\) \(\chi_{647}(607,\cdot)\)

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: \(\Q(\zeta_{17})\)
Fixed field: 17.17.942906198449660107953222334097149309547713921.1

Values on generators

\(5\) → \(e\left(\frac{12}{17}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{9}{17}\right)\)\(e\left(\frac{6}{17}\right)\)\(e\left(\frac{1}{17}\right)\)\(e\left(\frac{12}{17}\right)\)\(e\left(\frac{15}{17}\right)\)\(e\left(\frac{7}{17}\right)\)\(e\left(\frac{10}{17}\right)\)\(e\left(\frac{12}{17}\right)\)\(e\left(\frac{4}{17}\right)\)\(e\left(\frac{5}{17}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 647 }(53,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{647}(53,\cdot)) = \sum_{r\in \Z/647\Z} \chi_{647}(53,r) e\left(\frac{2r}{647}\right) = 11.0156951381+22.9271555284i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 647 }(53,·),\chi_{ 647 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{647}(53,\cdot),\chi_{647}(1,\cdot)) = \sum_{r\in \Z/647\Z} \chi_{647}(53,r) \chi_{647}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 647 }(53,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{647}(53,·)) = \sum_{r \in \Z/647\Z} \chi_{647}(53,r) e\left(\frac{1 r + 2 r^{-1}}{647}\right) = -1.3973631922+15.0799504325i \)