Properties

Label 953.16
Modulus $953$
Conductor $953$
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(953, base_ring=CyclotomicField(34))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([18]))
 
pari: [g,chi] = znchar(Mod(16,953))
 

Basic properties

Modulus: \(953\)
Conductor: \(953\)
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 953.g

\(\chi_{953}(16,\cdot)\) \(\chi_{953}(134,\cdot)\) \(\chi_{953}(238,\cdot)\) \(\chi_{953}(256,\cdot)\) \(\chi_{953}(276,\cdot)\) \(\chi_{953}(284,\cdot)\) \(\chi_{953}(417,\cdot)\) \(\chi_{953}(443,\cdot)\) \(\chi_{953}(604,\cdot)\) \(\chi_{953}(732,\cdot)\) \(\chi_{953}(770,\cdot)\) \(\chi_{953}(802,\cdot)\) \(\chi_{953}(882,\cdot)\) \(\chi_{953}(884,\cdot)\) \(\chi_{953}(889,\cdot)\) \(\chi_{953}(949,\cdot)\)

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: 17.17.462899178676311160288470191817767102492054133121.1

Values on generators

\(3\) → \(e\left(\frac{9}{17}\right)\)

Values

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

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 953 }(16,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{953}(16,\cdot)) = \sum_{r\in \Z/953\Z} \chi_{953}(16,r) e\left(\frac{2r}{953}\right) = 16.6216299016+26.0138697509i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 953 }(16,·),\chi_{ 953 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{953}(16,\cdot),\chi_{953}(1,\cdot)) = \sum_{r\in \Z/953\Z} \chi_{953}(16,r) \chi_{953}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 953 }(16,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{953}(16,·)) = \sum_{r \in \Z/953\Z} \chi_{953}(16,r) e\left(\frac{1 r + 2 r^{-1}}{953}\right) = 28.7735313819+-38.1023133194i \)