Properties

Conductor 101
Order 50
Real No
Primitive Yes
Parity Even
Orbit Label 101.h

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[20]
pari: [g,chi] = znchar(Mod(20,101))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 101
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 50
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.h
Orbit index = 8

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 ]

\(\chi_{101}(4,\cdot)\) \(\chi_{101}(9,\cdot)\) \(\chi_{101}(13,\cdot)\) \(\chi_{101}(20,\cdot)\) \(\chi_{101}(21,\cdot)\) \(\chi_{101}(22,\cdot)\) \(\chi_{101}(23,\cdot)\) \(\chi_{101}(30,\cdot)\) \(\chi_{101}(33,\cdot)\) \(\chi_{101}(43,\cdot)\) \(\chi_{101}(45,\cdot)\) \(\chi_{101}(47,\cdot)\) \(\chi_{101}(49,\cdot)\) \(\chi_{101}(64,\cdot)\) \(\chi_{101}(70,\cdot)\) \(\chi_{101}(76,\cdot)\) \(\chi_{101}(77,\cdot)\) \(\chi_{101}(82,\cdot)\) \(\chi_{101}(85,\cdot)\) \(\chi_{101}(96,\cdot)\)

Values on generators

\(2\) → \(e\left(\frac{13}{50}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{13}{50}\right)\)\(e\left(\frac{47}{50}\right)\)\(e\left(\frac{13}{25}\right)\)\(e\left(\frac{6}{25}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{17}{50}\right)\)\(e\left(\frac{39}{50}\right)\)\(e\left(\frac{22}{25}\right)\)\(-1\)\(e\left(\frac{19}{50}\right)\)
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)
\( \tau_{ a }( \chi_{ 101 }(20,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{101}(20,\cdot)) = \sum_{r\in \Z/101\Z} \chi_{101}(20,r) e\left(\frac{2r}{101}\right) = 9.9082179459+1.6814330602i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 101 }(20,·),\chi_{ 101 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{101}(20,\cdot),\chi_{101}(1,\cdot)) = \sum_{r\in \Z/101\Z} \chi_{101}(20,r) \chi_{101}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 101 }(20,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{101}(20,·)) = \sum_{r \in \Z/101\Z} \chi_{101}(20,r) e\left(\frac{1 r + 2 r^{-1}}{101}\right) = -11.0250018937+-11.7404345561i \)