Properties

Label 287.248
Modulus $287$
Conductor $287$
Order $60$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(287, base_ring=CyclotomicField(60))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([10,39]))
 
pari: [g,chi] = znchar(Mod(248,287))
 

Basic properties

Modulus: \(287\)
Conductor: \(287\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
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 287.bd

\(\chi_{287}(5,\cdot)\) \(\chi_{287}(33,\cdot)\) \(\chi_{287}(61,\cdot)\) \(\chi_{287}(80,\cdot)\) \(\chi_{287}(87,\cdot)\) \(\chi_{287}(103,\cdot)\) \(\chi_{287}(115,\cdot)\) \(\chi_{287}(131,\cdot)\) \(\chi_{287}(143,\cdot)\) \(\chi_{287}(159,\cdot)\) \(\chi_{287}(166,\cdot)\) \(\chi_{287}(185,\cdot)\) \(\chi_{287}(213,\cdot)\) \(\chi_{287}(241,\cdot)\) \(\chi_{287}(248,\cdot)\) \(\chi_{287}(285,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((206,211)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{13}{20}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\(-1\)\(1\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{37}{60}\right)\)\(e\left(\frac{23}{60}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 287 }(248,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 287 }(248,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 287 }(248,·),\chi_{ 287 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 287 }(248,·)) \;\) at \(\; a,b = \) e.g. 1,2