Properties

Label 675.38
Modulus $675$
Conductor $675$
Order $180$
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(180))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([130,171]))
 
pari: [g,chi] = znchar(Mod(38,675))
 

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: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 675.bi

\(\chi_{675}(2,\cdot)\) \(\chi_{675}(23,\cdot)\) \(\chi_{675}(38,\cdot)\) \(\chi_{675}(47,\cdot)\) \(\chi_{675}(77,\cdot)\) \(\chi_{675}(83,\cdot)\) \(\chi_{675}(92,\cdot)\) \(\chi_{675}(113,\cdot)\) \(\chi_{675}(122,\cdot)\) \(\chi_{675}(128,\cdot)\) \(\chi_{675}(137,\cdot)\) \(\chi_{675}(158,\cdot)\) \(\chi_{675}(167,\cdot)\) \(\chi_{675}(173,\cdot)\) \(\chi_{675}(203,\cdot)\) \(\chi_{675}(212,\cdot)\) \(\chi_{675}(227,\cdot)\) \(\chi_{675}(248,\cdot)\) \(\chi_{675}(263,\cdot)\) \(\chi_{675}(272,\cdot)\) \(\chi_{675}(302,\cdot)\) \(\chi_{675}(308,\cdot)\) \(\chi_{675}(317,\cdot)\) \(\chi_{675}(338,\cdot)\) \(\chi_{675}(347,\cdot)\) \(\chi_{675}(353,\cdot)\) \(\chi_{675}(362,\cdot)\) \(\chi_{675}(383,\cdot)\) \(\chi_{675}(392,\cdot)\) \(\chi_{675}(398,\cdot)\) ...

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

\((326,352)\) → \((e\left(\frac{13}{18}\right),e\left(\frac{19}{20}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 675 }(38, a) \) \(1\)\(1\)\(e\left(\frac{121}{180}\right)\)\(e\left(\frac{31}{90}\right)\)\(e\left(\frac{11}{36}\right)\)\(e\left(\frac{1}{60}\right)\)\(e\left(\frac{53}{90}\right)\)\(e\left(\frac{149}{180}\right)\)\(e\left(\frac{44}{45}\right)\)\(e\left(\frac{31}{45}\right)\)\(e\left(\frac{11}{60}\right)\)\(e\left(\frac{23}{30}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 675 }(38,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 675 }(38,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 675 }(38,·),\chi_{ 675 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 675 }(38,·)) \;\) at \(\; a,b = \) e.g. 1,2