Properties

Label 107.24
Modulus $107$
Conductor $107$
Order $106$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(107, base_ring=CyclotomicField(106))
 
M = H._module
 
chi = DirichletCharacter(H, M([73]))
 
pari: [g,chi] = znchar(Mod(24,107))
 

Basic properties

Modulus: \(107\)
Conductor: \(107\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(106\)
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 107.d

\(\chi_{107}(2,\cdot)\) \(\chi_{107}(5,\cdot)\) \(\chi_{107}(6,\cdot)\) \(\chi_{107}(7,\cdot)\) \(\chi_{107}(8,\cdot)\) \(\chi_{107}(15,\cdot)\) \(\chi_{107}(17,\cdot)\) \(\chi_{107}(18,\cdot)\) \(\chi_{107}(20,\cdot)\) \(\chi_{107}(21,\cdot)\) \(\chi_{107}(22,\cdot)\) \(\chi_{107}(24,\cdot)\) \(\chi_{107}(26,\cdot)\) \(\chi_{107}(28,\cdot)\) \(\chi_{107}(31,\cdot)\) \(\chi_{107}(32,\cdot)\) \(\chi_{107}(38,\cdot)\) \(\chi_{107}(43,\cdot)\) \(\chi_{107}(45,\cdot)\) \(\chi_{107}(46,\cdot)\) \(\chi_{107}(50,\cdot)\) \(\chi_{107}(51,\cdot)\) \(\chi_{107}(54,\cdot)\) \(\chi_{107}(55,\cdot)\) \(\chi_{107}(58,\cdot)\) \(\chi_{107}(59,\cdot)\) \(\chi_{107}(60,\cdot)\) \(\chi_{107}(63,\cdot)\) \(\chi_{107}(65,\cdot)\) \(\chi_{107}(66,\cdot)\) ...

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: $\Q(\zeta_{53})$
Fixed field: Number field defined by a degree 106 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{73}{106}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 107 }(24, a) \) \(-1\)\(1\)\(e\left(\frac{73}{106}\right)\)\(e\left(\frac{11}{53}\right)\)\(e\left(\frac{20}{53}\right)\)\(e\left(\frac{39}{106}\right)\)\(e\left(\frac{95}{106}\right)\)\(e\left(\frac{65}{106}\right)\)\(e\left(\frac{7}{106}\right)\)\(e\left(\frac{22}{53}\right)\)\(e\left(\frac{3}{53}\right)\)\(e\left(\frac{8}{53}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 107 }(24,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 107 }(24,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 107 }(24,·),\chi_{ 107 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 107 }(24,·)) \;\) at \(\; a,b = \) e.g. 1,2