Properties

Label 675.11
Modulus $675$
Conductor $675$
Order $90$
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(90))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([65,72]))
 
pari: [g,chi] = znchar(Mod(11,675))
 

Basic properties

Modulus: \(675\)
Conductor: \(675\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(90\)
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.bf

\(\chi_{675}(11,\cdot)\) \(\chi_{675}(41,\cdot)\) \(\chi_{675}(56,\cdot)\) \(\chi_{675}(86,\cdot)\) \(\chi_{675}(131,\cdot)\) \(\chi_{675}(146,\cdot)\) \(\chi_{675}(191,\cdot)\) \(\chi_{675}(221,\cdot)\) \(\chi_{675}(236,\cdot)\) \(\chi_{675}(266,\cdot)\) \(\chi_{675}(281,\cdot)\) \(\chi_{675}(311,\cdot)\) \(\chi_{675}(356,\cdot)\) \(\chi_{675}(371,\cdot)\) \(\chi_{675}(416,\cdot)\) \(\chi_{675}(446,\cdot)\) \(\chi_{675}(461,\cdot)\) \(\chi_{675}(491,\cdot)\) \(\chi_{675}(506,\cdot)\) \(\chi_{675}(536,\cdot)\) \(\chi_{675}(581,\cdot)\) \(\chi_{675}(596,\cdot)\) \(\chi_{675}(641,\cdot)\) \(\chi_{675}(671,\cdot)\)

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: Number field defined by a degree 90 polynomial

Values on generators

\((326,352)\) → \((e\left(\frac{13}{18}\right),e\left(\frac{4}{5}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 675 }(11, a) \) \(-1\)\(1\)\(e\left(\frac{47}{90}\right)\)\(e\left(\frac{2}{45}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{17}{90}\right)\)\(e\left(\frac{44}{45}\right)\)\(e\left(\frac{7}{90}\right)\)\(e\left(\frac{4}{45}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{1}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 675 }(11,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 675 }(11,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 675 }(11,·),\chi_{ 675 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 675 }(11,·)) \;\) at \(\; a,b = \) e.g. 1,2