Properties

Label 145.76
Modulus $145$
Conductor $29$
Order $28$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(145, base_ring=CyclotomicField(28))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,11]))
 
pari: [g,chi] = znchar(Mod(76,145))
 

Basic properties

Modulus: \(145\)
Conductor: \(29\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(28\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{29}(18,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 145.r

\(\chi_{145}(11,\cdot)\) \(\chi_{145}(21,\cdot)\) \(\chi_{145}(26,\cdot)\) \(\chi_{145}(31,\cdot)\) \(\chi_{145}(56,\cdot)\) \(\chi_{145}(61,\cdot)\) \(\chi_{145}(66,\cdot)\) \(\chi_{145}(76,\cdot)\) \(\chi_{145}(101,\cdot)\) \(\chi_{145}(106,\cdot)\) \(\chi_{145}(126,\cdot)\) \(\chi_{145}(131,\cdot)\)

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: \(\Q(\zeta_{28})\)
Fixed field: \(\Q(\zeta_{29})\)

Values on generators

\((117,31)\) → \((1,e\left(\frac{11}{28}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\(-1\)\(1\)\(e\left(\frac{11}{28}\right)\)\(e\left(\frac{27}{28}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{5}{28}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{23}{28}\right)\)\(-i\)\(e\left(\frac{1}{14}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 145 }(76,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 145 }(76,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 145 }(76,·),\chi_{ 145 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 145 }(76,·)) \;\) at \(\; a,b = \) e.g. 1,2