Properties

Label 675.53
Modulus $675$
Conductor $75$
Order $20$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(675, base_ring=CyclotomicField(20))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([10,7]))
 
pari: [g,chi] = znchar(Mod(53,675))
 

Basic properties

Modulus: \(675\)
Conductor: \(75\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{75}(53,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 675.w

\(\chi_{675}(53,\cdot)\) \(\chi_{675}(188,\cdot)\) \(\chi_{675}(242,\cdot)\) \(\chi_{675}(323,\cdot)\) \(\chi_{675}(377,\cdot)\) \(\chi_{675}(458,\cdot)\) \(\chi_{675}(512,\cdot)\) \(\chi_{675}(647,\cdot)\)

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: \(\Q(\zeta_{20})\)
Fixed field: \(\Q(\zeta_{75})^+\)

Values on generators

\((326,352)\) → \((-1,e\left(\frac{7}{20}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 675 }(53, a) \) \(1\)\(1\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{7}{10}\right)\)\(-i\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{3}{10}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 675 }(53,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 675 }(53,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 675 }(53,·),\chi_{ 675 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 675 }(53,·)) \;\) at \(\; a,b = \) e.g. 1,2