Properties

Conductor 97
Order 96
Real No
Primitive Yes
Parity Odd
Orbit Label 97.l

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[41]
 
pari: [g,chi] = znchar(Mod(41,97))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 97
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 96
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 = Odd
Orbit label = 97.l
Orbit index = 12

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 ]
 

\(\chi_{97}(5,\cdot)\) \(\chi_{97}(7,\cdot)\) \(\chi_{97}(10,\cdot)\) \(\chi_{97}(13,\cdot)\) \(\chi_{97}(14,\cdot)\) \(\chi_{97}(15,\cdot)\) \(\chi_{97}(17,\cdot)\) \(\chi_{97}(21,\cdot)\) \(\chi_{97}(23,\cdot)\) \(\chi_{97}(26,\cdot)\) \(\chi_{97}(29,\cdot)\) \(\chi_{97}(37,\cdot)\) \(\chi_{97}(38,\cdot)\) \(\chi_{97}(39,\cdot)\) \(\chi_{97}(40,\cdot)\) \(\chi_{97}(41,\cdot)\) \(\chi_{97}(56,\cdot)\) \(\chi_{97}(57,\cdot)\) \(\chi_{97}(58,\cdot)\) \(\chi_{97}(59,\cdot)\) \(\chi_{97}(60,\cdot)\) \(\chi_{97}(68,\cdot)\) \(\chi_{97}(71,\cdot)\) \(\chi_{97}(74,\cdot)\) \(\chi_{97}(76,\cdot)\) \(\chi_{97}(80,\cdot)\) \(\chi_{97}(82,\cdot)\) \(\chi_{97}(83,\cdot)\) \(\chi_{97}(84,\cdot)\) \(\chi_{97}(87,\cdot)\) ...

Values on generators

\(5\) → \(e\left(\frac{85}{96}\right)\)

Values

-11234567891011
\(-1\)\(1\)\(e\left(\frac{5}{48}\right)\)\(e\left(\frac{47}{48}\right)\)\(e\left(\frac{5}{24}\right)\)\(e\left(\frac{85}{96}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{43}{96}\right)\)\(e\left(\frac{5}{16}\right)\)\(e\left(\frac{23}{24}\right)\)\(e\left(\frac{95}{96}\right)\)\(e\left(\frac{7}{48}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{96})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 97 }(41,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{97}(41,\cdot)) = \sum_{r\in \Z/97\Z} \chi_{97}(41,r) e\left(\frac{2r}{97}\right) = -2.7710555277+9.4509920782i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 97 }(41,·),\chi_{ 97 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{97}(41,\cdot),\chi_{97}(1,\cdot)) = \sum_{r\in \Z/97\Z} \chi_{97}(41,r) \chi_{97}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 97 }(41,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{97}(41,·)) = \sum_{r \in \Z/97\Z} \chi_{97}(41,r) e\left(\frac{1 r + 2 r^{-1}}{97}\right) = 3.119368003+-9.1893618112i \)