Properties

Label 101.59
Modulus $101$
Conductor $101$
Order $100$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(101, base_ring=CyclotomicField(100))
 
M = H._module
 
chi = DirichletCharacter(H, M([29]))
 
pari: [g,chi] = znchar(Mod(59,101))
 

Basic properties

Modulus: \(101\)
Conductor: \(101\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(100\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 101.i

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

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: $\Q(\zeta_{100})$
Fixed field: Number field defined by a degree 100 polynomial

Values on generators

\(2\) → \(e\left(\frac{29}{100}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 101 }(59, a) \) \(-1\)\(1\)\(e\left(\frac{29}{100}\right)\)\(e\left(\frac{1}{100}\right)\)\(e\left(\frac{29}{50}\right)\)\(e\left(\frac{24}{25}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{61}{100}\right)\)\(e\left(\frac{87}{100}\right)\)\(e\left(\frac{1}{50}\right)\)\(i\)\(e\left(\frac{77}{100}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 101 }(59,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 101 }(59,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 101 }(59,·),\chi_{ 101 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 101 }(59,·)) \;\) at \(\; a,b = \) e.g. 1,2