Properties

Label 101.25
Modulus $101$
Conductor $101$
Order $25$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(101, base_ring=CyclotomicField(50))
 
M = H._module
 
chi = DirichletCharacter(H, M([24]))
 
pari: [g,chi] = znchar(Mod(25,101))
 

Basic properties

Modulus: \(101\)
Conductor: \(101\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(25\)
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.g

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

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

Related number fields

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

Values on generators

\(2\) → \(e\left(\frac{12}{25}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 101 }(25, a) \) \(1\)\(1\)\(e\left(\frac{12}{25}\right)\)\(e\left(\frac{3}{25}\right)\)\(e\left(\frac{24}{25}\right)\)\(e\left(\frac{13}{25}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{8}{25}\right)\)\(e\left(\frac{11}{25}\right)\)\(e\left(\frac{6}{25}\right)\)\(1\)\(e\left(\frac{6}{25}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 101 }(25,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 101 }(25,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 101 }(25,·),\chi_{ 101 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 101 }(25,·)) \;\) at \(\; a,b = \) e.g. 1,2