Properties

Label 847.39
Modulus 847847
Conductor 847847
Order 330330
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(847, base_ring=CyclotomicField(330)) M = H._module chi = DirichletCharacter(H, M([220,237]))
 
Copy content pari:[g,chi] = znchar(Mod(39,847))
 

Basic properties

Modulus: 847847
Conductor: 847847
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: 330330
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: yes
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 847.bf

χ847(2,)\chi_{847}(2,\cdot) χ847(18,)\chi_{847}(18,\cdot) χ847(30,)\chi_{847}(30,\cdot) χ847(39,)\chi_{847}(39,\cdot) χ847(46,)\chi_{847}(46,\cdot) χ847(51,)\chi_{847}(51,\cdot) χ847(72,)\chi_{847}(72,\cdot) χ847(74,)\chi_{847}(74,\cdot) χ847(79,)\chi_{847}(79,\cdot) χ847(95,)\chi_{847}(95,\cdot) χ847(107,)\chi_{847}(107,\cdot) χ847(116,)\chi_{847}(116,\cdot) χ847(123,)\chi_{847}(123,\cdot) χ847(128,)\chi_{847}(128,\cdot) χ847(149,)\chi_{847}(149,\cdot) χ847(151,)\chi_{847}(151,\cdot) χ847(156,)\chi_{847}(156,\cdot) χ847(172,)\chi_{847}(172,\cdot) χ847(184,)\chi_{847}(184,\cdot) χ847(193,)\chi_{847}(193,\cdot) χ847(200,)\chi_{847}(200,\cdot) χ847(205,)\chi_{847}(205,\cdot) χ847(226,)\chi_{847}(226,\cdot) χ847(228,)\chi_{847}(228,\cdot) χ847(249,)\chi_{847}(249,\cdot) χ847(261,)\chi_{847}(261,\cdot) χ847(270,)\chi_{847}(270,\cdot) χ847(277,)\chi_{847}(277,\cdot) χ847(303,)\chi_{847}(303,\cdot) χ847(305,)\chi_{847}(305,\cdot) ...

Copy content sage:chi.galois_orbit()
 
Copy content pari:order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: Q(ζ165)\Q(\zeta_{165})
Fixed field: Number field defined by a degree 330 polynomial (not computed)

Values on generators

(122,365)(122,365)(e(23),e(79110))(e\left(\frac{2}{3}\right),e\left(\frac{79}{110}\right))

First values

aa 1-11122334455668899101012121313
χ847(39,a) \chi_{ 847 }(39, a) 1-111e(17330)e\left(\frac{17}{330}\right)e(1315)e\left(\frac{13}{15}\right)e(17165)e\left(\frac{17}{165}\right)e(79165)e\left(\frac{79}{165}\right)e(101110)e\left(\frac{101}{110}\right)e(17110)e\left(\frac{17}{110}\right)e(1115)e\left(\frac{11}{15}\right)e(3566)e\left(\frac{35}{66}\right)e(3233)e\left(\frac{32}{33}\right)e(59110)e\left(\frac{59}{110}\right)
Copy content sage:chi.jacobi_sum(n)
 
χ847(39,a)   \chi_{ 847 }(39,a) \; at   a=\;a = e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
τa(χ847(39,))   \tau_{ a }( \chi_{ 847 }(39,·) )\; at   a=\;a = e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
J(χ847(39,),χ847(n,))   J(\chi_{ 847 }(39,·),\chi_{ 847 }(n,·)) \; for   n= \; n = e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
K(a,b,χ847(39,))  K(a,b,\chi_{ 847 }(39,·)) \; at   a,b=\; a,b = e.g. 1,2