Properties

Label 919.703
Modulus $919$
Conductor $919$
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(919, base_ring=CyclotomicField(34))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([2]))
 
pari: [g,chi] = znchar(Mod(703,919))
 

Basic properties

Modulus: \(919\)
Conductor: \(919\)
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 919.f

\(\chi_{919}(58,\cdot)\) \(\chi_{919}(162,\cdot)\) \(\chi_{919}(206,\cdot)\) \(\chi_{919}(229,\cdot)\) \(\chi_{919}(234,\cdot)\) \(\chi_{919}(284,\cdot)\) \(\chi_{919}(288,\cdot)\) \(\chi_{919}(305,\cdot)\) \(\chi_{919}(338,\cdot)\) \(\chi_{919}(416,\cdot)\) \(\chi_{919}(512,\cdot)\) \(\chi_{919}(535,\cdot)\) \(\chi_{919}(607,\cdot)\) \(\chi_{919}(703,\cdot)\) \(\chi_{919}(706,\cdot)\) \(\chi_{919}(849,\cdot)\)

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: 17.17.258850007664814362506653464185428842369437873281.1

Values on generators

\(7\) → \(e\left(\frac{1}{17}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{13}{17}\right)\)\(e\left(\frac{5}{17}\right)\)\(e\left(\frac{9}{17}\right)\)\(e\left(\frac{5}{17}\right)\)\(e\left(\frac{1}{17}\right)\)\(e\left(\frac{1}{17}\right)\)\(e\left(\frac{5}{17}\right)\)\(e\left(\frac{10}{17}\right)\)\(e\left(\frac{1}{17}\right)\)\(e\left(\frac{7}{17}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 919 }(703,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{919}(703,\cdot)) = \sum_{r\in \Z/919\Z} \chi_{919}(703,r) e\left(\frac{2r}{919}\right) = 9.4812682397+-28.7941930355i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 919 }(703,·),\chi_{ 919 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{919}(703,\cdot),\chi_{919}(1,\cdot)) = \sum_{r\in \Z/919\Z} \chi_{919}(703,r) \chi_{919}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 919 }(703,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{919}(703,·)) = \sum_{r \in \Z/919\Z} \chi_{919}(703,r) e\left(\frac{1 r + 2 r^{-1}}{919}\right) = 26.0015731153+-23.7035658536i \)