Properties

Label 675.52
Modulus $675$
Conductor $675$
Order $180$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(675, base_ring=CyclotomicField(180))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([100,9]))
 
pari: [g,chi] = znchar(Mod(52,675))
 

Basic properties

Modulus: \(675\)
Conductor: \(675\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(180\)
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 675.bj

\(\chi_{675}(13,\cdot)\) \(\chi_{675}(22,\cdot)\) \(\chi_{675}(52,\cdot)\) \(\chi_{675}(58,\cdot)\) \(\chi_{675}(67,\cdot)\) \(\chi_{675}(88,\cdot)\) \(\chi_{675}(97,\cdot)\) \(\chi_{675}(103,\cdot)\) \(\chi_{675}(112,\cdot)\) \(\chi_{675}(133,\cdot)\) \(\chi_{675}(142,\cdot)\) \(\chi_{675}(148,\cdot)\) \(\chi_{675}(178,\cdot)\) \(\chi_{675}(187,\cdot)\) \(\chi_{675}(202,\cdot)\) \(\chi_{675}(223,\cdot)\) \(\chi_{675}(238,\cdot)\) \(\chi_{675}(247,\cdot)\) \(\chi_{675}(277,\cdot)\) \(\chi_{675}(283,\cdot)\) \(\chi_{675}(292,\cdot)\) \(\chi_{675}(313,\cdot)\) \(\chi_{675}(322,\cdot)\) \(\chi_{675}(328,\cdot)\) \(\chi_{675}(337,\cdot)\) \(\chi_{675}(358,\cdot)\) \(\chi_{675}(367,\cdot)\) \(\chi_{675}(373,\cdot)\) \(\chi_{675}(403,\cdot)\) \(\chi_{675}(412,\cdot)\) ...

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: $\Q(\zeta_{180})$
Fixed field: Number field defined by a degree 180 polynomial (not computed)

Values on generators

\((326,352)\) → \((e\left(\frac{5}{9}\right),e\left(\frac{1}{20}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 675 }(52, a) \) \(-1\)\(1\)\(e\left(\frac{109}{180}\right)\)\(e\left(\frac{19}{90}\right)\)\(e\left(\frac{5}{36}\right)\)\(e\left(\frac{49}{60}\right)\)\(e\left(\frac{1}{45}\right)\)\(e\left(\frac{71}{180}\right)\)\(e\left(\frac{67}{90}\right)\)\(e\left(\frac{19}{45}\right)\)\(e\left(\frac{59}{60}\right)\)\(e\left(\frac{17}{30}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 675 }(52,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 675 }(52,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 675 }(52,·),\chi_{ 675 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 675 }(52,·)) \;\) at \(\; a,b = \) e.g. 1,2