Properties

Conductor 101
Order 25
Real No
Primitive Yes
Parity Even
Orbit Label 101.g

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[19]
pari: [g,chi] = znchar(Mod(19,101))

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
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

\(2\) → \(e\left(\frac{24}{25}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{24}{25}\right)\)\(e\left(\frac{6}{25}\right)\)\(e\left(\frac{23}{25}\right)\)\(e\left(\frac{1}{25}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{16}{25}\right)\)\(e\left(\frac{22}{25}\right)\)\(e\left(\frac{12}{25}\right)\)\(1\)\(e\left(\frac{12}{25}\right)\)
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)
\( \tau_{ a }( \chi_{ 101 }(19,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{101}(19,\cdot)) = \sum_{r\in \Z/101\Z} \chi_{101}(19,r) e\left(\frac{2r}{101}\right) = 8.5622408996+5.2619417308i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 101 }(19,·),\chi_{ 101 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{101}(19,\cdot),\chi_{101}(1,\cdot)) = \sum_{r\in \Z/101\Z} \chi_{101}(19,r) \chi_{101}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 101 }(19,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{101}(19,·)) = \sum_{r \in \Z/101\Z} \chi_{101}(19,r) e\left(\frac{1 r + 2 r^{-1}}{101}\right) = 2.8112354467+-0.3551416266i \)