Properties

Label 675.16
Modulus $675$
Conductor $675$
Order $45$
Real no
Primitive yes
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(90))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([20,18]))
 
pari: [g,chi] = znchar(Mod(16,675))
 

Basic properties

Modulus: \(675\)
Conductor: \(675\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(45\)
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 675.bc

\(\chi_{675}(16,\cdot)\) \(\chi_{675}(31,\cdot)\) \(\chi_{675}(61,\cdot)\) \(\chi_{675}(106,\cdot)\) \(\chi_{675}(121,\cdot)\) \(\chi_{675}(166,\cdot)\) \(\chi_{675}(196,\cdot)\) \(\chi_{675}(211,\cdot)\) \(\chi_{675}(241,\cdot)\) \(\chi_{675}(256,\cdot)\) \(\chi_{675}(286,\cdot)\) \(\chi_{675}(331,\cdot)\) \(\chi_{675}(346,\cdot)\) \(\chi_{675}(391,\cdot)\) \(\chi_{675}(421,\cdot)\) \(\chi_{675}(436,\cdot)\) \(\chi_{675}(466,\cdot)\) \(\chi_{675}(481,\cdot)\) \(\chi_{675}(511,\cdot)\) \(\chi_{675}(556,\cdot)\) \(\chi_{675}(571,\cdot)\) \(\chi_{675}(616,\cdot)\) \(\chi_{675}(646,\cdot)\) \(\chi_{675}(661,\cdot)\)

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: $\Q(\zeta_{45})$
Fixed field: 45.45.6444338306249279600681470320699578378786756892621858688031629726344906572421677992679178714752197265625.1

Values on generators

\((326,352)\) → \((e\left(\frac{2}{9}\right),e\left(\frac{1}{5}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 675 }(16, a) \) \(1\)\(1\)\(e\left(\frac{19}{45}\right)\)\(e\left(\frac{38}{45}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{4}{45}\right)\)\(e\left(\frac{26}{45}\right)\)\(e\left(\frac{44}{45}\right)\)\(e\left(\frac{31}{45}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{4}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 675 }(16,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 675 }(16,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 675 }(16,·),\chi_{ 675 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 675 }(16,·)) \;\) at \(\; a,b = \) e.g. 1,2