Properties

Label 101.22
Modulus $101$
Conductor $101$
Order $50$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(101)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([7]))
 
pari: [g,chi] = znchar(Mod(22,101))
 

Basic properties

Modulus: \(101\)
Conductor: \(101\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(50\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 101.h

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

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Values on generators

\(2\) → \(e\left(\frac{7}{50}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{7}{50}\right)\)\(e\left(\frac{33}{50}\right)\)\(e\left(\frac{7}{25}\right)\)\(e\left(\frac{9}{25}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{13}{50}\right)\)\(e\left(\frac{21}{50}\right)\)\(e\left(\frac{8}{25}\right)\)\(-1\)\(e\left(\frac{41}{50}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{25})\)
Fixed field: Number field defined by a degree 50 polynomial

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 101 }(22,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{101}(22,\cdot)) = \sum_{r\in \Z/101\Z} \chi_{101}(22,r) e\left(\frac{2r}{101}\right) = 4.0325018176+9.2053750109i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 101 }(22,·),\chi_{ 101 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{101}(22,\cdot),\chi_{101}(1,\cdot)) = \sum_{r\in \Z/101\Z} \chi_{101}(22,r) \chi_{101}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 101 }(22,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{101}(22,·)) = \sum_{r \in \Z/101\Z} \chi_{101}(22,r) e\left(\frac{1 r + 2 r^{-1}}{101}\right) = -5.726530766+-2.6947008337i \)