Properties

Label 34.7
Modulus $34$
Conductor $17$
Order $16$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(34, base_ring=CyclotomicField(16))
 
M = H._module
 
chi = DirichletCharacter(H, M([11]))
 
pari: [g,chi] = znchar(Mod(7,34))
 

Basic properties

Modulus: \(34\)
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 34.e

\(\chi_{34}(3,\cdot)\) \(\chi_{34}(5,\cdot)\) \(\chi_{34}(7,\cdot)\) \(\chi_{34}(11,\cdot)\) \(\chi_{34}(23,\cdot)\) \(\chi_{34}(27,\cdot)\) \(\chi_{34}(29,\cdot)\) \(\chi_{34}(31,\cdot)\)

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

Related number fields

Field of values: \(\Q(\zeta_{16})\)
Fixed field: Number field defined by a degree 16 polynomial

Values on generators

\(3\) → \(e\left(\frac{11}{16}\right)\)

Values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(19\)\(21\)\(23\)
\( \chi_{ 34 }(7, a) \) \(-1\)\(1\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{13}{16}\right)\)\(-i\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{5}{8}\right)\)\(i\)\(e\left(\frac{5}{16}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 34 }(7,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 34 }(7,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 34 }(7,·),\chi_{ 34 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 34 }(7,·)) \;\) at \(\; a,b = \) e.g. 1,2