Properties

Label 153.5
Modulus $153$
Conductor $153$
Order $48$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(153, base_ring=CyclotomicField(48))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([40,15]))
 
pari: [g,chi] = znchar(Mod(5,153))
 

Basic properties

Modulus: \(153\)
Conductor: \(153\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(48\)
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 153.s

\(\chi_{153}(5,\cdot)\) \(\chi_{153}(11,\cdot)\) \(\chi_{153}(14,\cdot)\) \(\chi_{153}(20,\cdot)\) \(\chi_{153}(23,\cdot)\) \(\chi_{153}(29,\cdot)\) \(\chi_{153}(41,\cdot)\) \(\chi_{153}(56,\cdot)\) \(\chi_{153}(65,\cdot)\) \(\chi_{153}(74,\cdot)\) \(\chi_{153}(92,\cdot)\) \(\chi_{153}(95,\cdot)\) \(\chi_{153}(113,\cdot)\) \(\chi_{153}(122,\cdot)\) \(\chi_{153}(131,\cdot)\) \(\chi_{153}(146,\cdot)\)

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

Related number fields

Field of values: \(\Q(\zeta_{48})\)
Fixed field: Number field defined by a degree 48 polynomial

Values on generators

\((137,37)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{5}{16}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 153 }(5, a) \) \(1\)\(1\)\(e\left(\frac{5}{24}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{35}{48}\right)\)\(e\left(\frac{37}{48}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{1}{48}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{47}{48}\right)\)\(e\left(\frac{5}{6}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 153 }(5,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 153 }(5,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 153 }(5,·),\chi_{ 153 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 153 }(5,·)) \;\) at \(\; a,b = \) e.g. 1,2