Properties

Conductor 101
Order 25
Real no
Primitive yes
Minimal yes
Parity even
Orbit label 101.g

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
 
sage: H = DirichletGroup_conrey(101)
 
sage: chi = H[5]
 
pari: [g,chi] = znchar(Mod(5,101))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 101
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 25
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 = 101.g
Orbit index = 7

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 ]
 

\(\chi_{101}(5,\cdot)\) \(\chi_{101}(16,\cdot)\) \(\chi_{101}(19,\cdot)\) \(\chi_{101}(24,\cdot)\) \(\chi_{101}(25,\cdot)\) \(\chi_{101}(31,\cdot)\) \(\chi_{101}(37,\cdot)\) \(\chi_{101}(52,\cdot)\) \(\chi_{101}(54,\cdot)\) \(\chi_{101}(56,\cdot)\) \(\chi_{101}(58,\cdot)\) \(\chi_{101}(68,\cdot)\) \(\chi_{101}(71,\cdot)\) \(\chi_{101}(78,\cdot)\) \(\chi_{101}(79,\cdot)\) \(\chi_{101}(80,\cdot)\) \(\chi_{101}(81,\cdot)\) \(\chi_{101}(88,\cdot)\) \(\chi_{101}(92,\cdot)\) \(\chi_{101}(97,\cdot)\)

Values on generators

\(2\) → \(e\left(\frac{6}{25}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{6}{25}\right)\)\(e\left(\frac{14}{25}\right)\)\(e\left(\frac{12}{25}\right)\)\(e\left(\frac{19}{25}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{4}{25}\right)\)\(e\left(\frac{18}{25}\right)\)\(e\left(\frac{3}{25}\right)\)\(1\)\(e\left(\frac{3}{25}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{25})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 101 }(5,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{101}(5,\cdot)) = \sum_{r\in \Z/101\Z} \chi_{101}(5,r) e\left(\frac{2r}{101}\right) = 7.8201713494+-6.3122832688i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 101 }(5,·),\chi_{ 101 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{101}(5,\cdot),\chi_{101}(1,\cdot)) = \sum_{r\in \Z/101\Z} \chi_{101}(5,r) \chi_{101}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 101 }(5,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{101}(5,·)) = \sum_{r \in \Z/101\Z} \chi_{101}(5,r) e\left(\frac{1 r + 2 r^{-1}}{101}\right) = 9.6822257435+9.0922151686i \)