Properties

Label 51.7
Modulus $51$
Conductor $17$
Order $16$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(51, base_ring=CyclotomicField(16))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,11]))
 
pari: [g,chi] = znchar(Mod(7,51))
 

Basic properties

Modulus: \(51\)
Conductor: \(17\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(16\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{17}(7,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 51.j

\(\chi_{51}(7,\cdot)\) \(\chi_{51}(10,\cdot)\) \(\chi_{51}(22,\cdot)\) \(\chi_{51}(28,\cdot)\) \(\chi_{51}(31,\cdot)\) \(\chi_{51}(37,\cdot)\) \(\chi_{51}(40,\cdot)\) \(\chi_{51}(46,\cdot)\)

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: \(\Q(\zeta_{16})\)
Fixed field: \(\Q(\zeta_{17})\)

Values on generators

\((35,37)\) → \((1,e\left(\frac{11}{16}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(-1\)\(1\)\(e\left(\frac{5}{8}\right)\)\(i\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{13}{16}\right)\)\(-i\)\(e\left(\frac{3}{16}\right)\)\(-1\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 51 }(7,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{51}(7,\cdot)) = \sum_{r\in \Z/51\Z} \chi_{51}(7,r) e\left(\frac{2r}{51}\right) = -0.6753510742+4.0674194432i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 51 }(7,·),\chi_{ 51 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{51}(7,\cdot),\chi_{51}(1,\cdot)) = \sum_{r\in \Z/51\Z} \chi_{51}(7,r) \chi_{51}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 51 }(7,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{51}(7,·)) = \sum_{r \in \Z/51\Z} \chi_{51}(7,r) e\left(\frac{1 r + 2 r^{-1}}{51}\right) = 5.9634387425+2.4701372055i \)