Properties

Modulus 97
Conductor 97
Order 8
Real no
Primitive yes
Minimal yes
Parity even
Orbit label 97.f

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(97)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([3]))
 
pari: [g,chi] = znchar(Mod(50,97))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 97
Conductor = 97
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 8
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = yes
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = even
Orbit label = 97.f
Orbit index = 6

Galois orbit

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

\(\chi_{97}(33,\cdot)\) \(\chi_{97}(47,\cdot)\) \(\chi_{97}(50,\cdot)\) \(\chi_{97}(64,\cdot)\)

Values on generators

\(5\) → \(e\left(\frac{3}{8}\right)\)

Values

-11234567891011
\(1\)\(1\)\(-i\)\(i\)\(-1\)\(e\left(\frac{3}{8}\right)\)\(1\)\(e\left(\frac{5}{8}\right)\)\(i\)\(-1\)\(e\left(\frac{1}{8}\right)\)\(i\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{8})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 97 }(50,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{97}(50,\cdot)) = \sum_{r\in \Z/97\Z} \chi_{97}(50,r) e\left(\frac{2r}{97}\right) = -9.720403695+-1.5854816321i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 97 }(50,·),\chi_{ 97 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{97}(50,\cdot),\chi_{97}(1,\cdot)) = \sum_{r\in \Z/97\Z} \chi_{97}(50,r) \chi_{97}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 97 }(50,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{97}(50,·)) = \sum_{r \in \Z/97\Z} \chi_{97}(50,r) e\left(\frac{1 r + 2 r^{-1}}{97}\right) = 9.611617533+-9.611617533i \)