Properties

Conductor 101
Order 100
Real No
Primitive Yes
Parity Odd
Orbit Label 101.i

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(101)
sage: chi = H[8]
pari: [g,chi] = znchar(Mod(8,101))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 101
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 100
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 = 101.i
Orbit index = 9

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 ]

\(\chi_{101}(2,\cdot)\) \(\chi_{101}(3,\cdot)\) \(\chi_{101}(7,\cdot)\) \(\chi_{101}(8,\cdot)\) \(\chi_{101}(11,\cdot)\) \(\chi_{101}(12,\cdot)\) \(\chi_{101}(15,\cdot)\) \(\chi_{101}(18,\cdot)\) \(\chi_{101}(26,\cdot)\) \(\chi_{101}(27,\cdot)\) \(\chi_{101}(28,\cdot)\) \(\chi_{101}(29,\cdot)\) \(\chi_{101}(34,\cdot)\) \(\chi_{101}(35,\cdot)\) \(\chi_{101}(38,\cdot)\) \(\chi_{101}(40,\cdot)\) \(\chi_{101}(42,\cdot)\) \(\chi_{101}(46,\cdot)\) \(\chi_{101}(48,\cdot)\) \(\chi_{101}(50,\cdot)\) \(\chi_{101}(51,\cdot)\) \(\chi_{101}(53,\cdot)\) \(\chi_{101}(55,\cdot)\) \(\chi_{101}(59,\cdot)\) \(\chi_{101}(61,\cdot)\) \(\chi_{101}(63,\cdot)\) \(\chi_{101}(66,\cdot)\) \(\chi_{101}(67,\cdot)\) \(\chi_{101}(72,\cdot)\) \(\chi_{101}(73,\cdot)\) ...

Values on generators

\(2\) → \(e\left(\frac{3}{100}\right)\)

Values

-11234567891011
\(-1\)\(1\)\(e\left(\frac{3}{100}\right)\)\(e\left(\frac{7}{100}\right)\)\(e\left(\frac{3}{50}\right)\)\(e\left(\frac{18}{25}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{27}{100}\right)\)\(e\left(\frac{9}{100}\right)\)\(e\left(\frac{7}{50}\right)\)\(-i\)\(e\left(\frac{39}{100}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{100})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 101 }(8,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{101}(8,\cdot)) = \sum_{r\in \Z/101\Z} \chi_{101}(8,r) e\left(\frac{2r}{101}\right) = -5.1882667345+8.6070836113i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 101 }(8,·),\chi_{ 101 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{101}(8,\cdot),\chi_{101}(1,\cdot)) = \sum_{r\in \Z/101\Z} \chi_{101}(8,r) \chi_{101}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 101 }(8,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{101}(8,·)) = \sum_{r \in \Z/101\Z} \chi_{101}(8,r) e\left(\frac{1 r + 2 r^{-1}}{101}\right) = 0.1069117699+-1.131008387i \)