Properties

Label 287.134
Modulus $287$
Conductor $41$
Order $40$
Real no
Primitive no
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(40))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,3]))
 
pari: [g,chi] = znchar(Mod(134,287))
 

Basic properties

Modulus: \(287\)
Conductor: \(41\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(40\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{41}(11,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 287.ba

\(\chi_{287}(15,\cdot)\) \(\chi_{287}(22,\cdot)\) \(\chi_{287}(29,\cdot)\) \(\chi_{287}(71,\cdot)\) \(\chi_{287}(99,\cdot)\) \(\chi_{287}(106,\cdot)\) \(\chi_{287}(134,\cdot)\) \(\chi_{287}(176,\cdot)\) \(\chi_{287}(183,\cdot)\) \(\chi_{287}(190,\cdot)\) \(\chi_{287}(211,\cdot)\) \(\chi_{287}(218,\cdot)\) \(\chi_{287}(239,\cdot)\) \(\chi_{287}(253,\cdot)\) \(\chi_{287}(274,\cdot)\) \(\chi_{287}(281,\cdot)\)

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: \(\Q(\zeta_{40})\)
Fixed field: \(\Q(\zeta_{41})\)

Values on generators

\((206,211)\) → \((1,e\left(\frac{3}{40}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\(-1\)\(1\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{3}{40}\right)\)\(e\left(\frac{17}{20}\right)\)\(i\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{9}{40}\right)\)\(e\left(\frac{1}{40}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 287 }(134,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 287 }(134,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 287 }(134,·),\chi_{ 287 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 287 }(134,·)) \;\) at \(\; a,b = \) e.g. 1,2