Properties

Conductor 97
Order 8
Real No
Primitive Yes
Parity Even
Orbit Label 97.f

Related objects

Learn more about

Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(97)
sage: chi = H[64]
pari: [g,chi] = znchar(Mod(64,97))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
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
sage: chi.is_odd()
pari: zncharisodd(g,chi)
Parity = Even
Orbit label = 97.f
Orbit index = 6

Galois orbit

sage: chi.sage_character().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

\(5\) → \(e\left(\frac{1}{8}\right)\)

Values

-11234567891011
\(1\)\(1\)\(i\)\(-i\)\(-1\)\(e\left(\frac{1}{8}\right)\)\(1\)\(e\left(\frac{7}{8}\right)\)\(-i\)\(-1\)\(e\left(\frac{3}{8}\right)\)\(-i\)
value at  e.g. 2

Related number fields

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

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 97 }(64,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{97}(64,\cdot)) = \sum_{r\in \Z/97\Z} \chi_{97}(64,r) e\left(\frac{2r}{97}\right) = 3.5688159399+-9.179518113i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 97 }(64,·),\chi_{ 97 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{97}(64,\cdot),\chi_{97}(1,\cdot)) = \sum_{r\in \Z/97\Z} \chi_{97}(64,r) \chi_{97}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 97 }(64,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{97}(64,·)) = \sum_{r \in \Z/97\Z} \chi_{97}(64,r) e\left(\frac{1 r + 2 r^{-1}}{97}\right) = -4.6685548813+-4.6685548813i \)