Properties

Label 97.19
Modulus $97$
Conductor $97$
Order $32$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(97, base_ring=CyclotomicField(32))
 
M = H._module
 
chi = DirichletCharacter(H, M([27]))
 
pari: [g,chi] = znchar(Mod(19,97))
 

Basic properties

Modulus: \(97\)
Conductor: \(97\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(32\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 97.j

\(\chi_{97}(19,\cdot)\) \(\chi_{97}(20,\cdot)\) \(\chi_{97}(28,\cdot)\) \(\chi_{97}(30,\cdot)\) \(\chi_{97}(34,\cdot)\) \(\chi_{97}(42,\cdot)\) \(\chi_{97}(45,\cdot)\) \(\chi_{97}(46,\cdot)\) \(\chi_{97}(51,\cdot)\) \(\chi_{97}(52,\cdot)\) \(\chi_{97}(55,\cdot)\) \(\chi_{97}(63,\cdot)\) \(\chi_{97}(67,\cdot)\) \(\chi_{97}(69,\cdot)\) \(\chi_{97}(77,\cdot)\) \(\chi_{97}(78,\cdot)\)

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: \(\Q(\zeta_{32})\)
Fixed field: Number field defined by a degree 32 polynomial

Values on generators

\(5\) → \(e\left(\frac{27}{32}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 97 }(19, a) \) \(-1\)\(1\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{27}{32}\right)\)\(-i\)\(e\left(\frac{5}{32}\right)\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{17}{32}\right)\)\(e\left(\frac{9}{16}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 97 }(19,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 97 }(19,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 97 }(19,·),\chi_{ 97 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 97 }(19,·)) \;\) at \(\; a,b = \) e.g. 1,2