Properties

Label 966.751
Modulus 966966
Conductor 161161
Order 6666
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(966, base_ring=CyclotomicField(66)) M = H._module chi = DirichletCharacter(H, M([0,22,51]))
 
Copy content pari:[g,chi] = znchar(Mod(751,966))
 

Basic properties

Modulus: 966966
Conductor: 161161
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: 6666
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from χ161(107,)\chi_{161}(107,\cdot)
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 966.z

χ966(37,)\chi_{966}(37,\cdot) χ966(67,)\chi_{966}(67,\cdot) χ966(79,)\chi_{966}(79,\cdot) χ966(109,)\chi_{966}(109,\cdot) χ966(205,)\chi_{966}(205,\cdot) χ966(235,)\chi_{966}(235,\cdot) χ966(247,)\chi_{966}(247,\cdot) χ966(319,)\chi_{966}(319,\cdot) χ966(373,)\chi_{966}(373,\cdot) χ966(457,)\chi_{966}(457,\cdot) χ966(571,)\chi_{966}(571,\cdot) χ966(613,)\chi_{966}(613,\cdot) χ966(655,)\chi_{966}(655,\cdot) χ966(697,)\chi_{966}(697,\cdot) χ966(709,)\chi_{966}(709,\cdot) χ966(751,)\chi_{966}(751,\cdot) χ966(793,)\chi_{966}(793,\cdot) χ966(835,)\chi_{966}(835,\cdot) χ966(865,)\chi_{966}(865,\cdot) χ966(907,)\chi_{966}(907,\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(ζ33)\Q(\zeta_{33})
Fixed field: Number field defined by a degree 66 polynomial

Values on generators

(323,829,925)(323,829,925)(1,e(13),e(1722))(1,e\left(\frac{1}{3}\right),e\left(\frac{17}{22}\right))

First values

aa 1-11155111113131717191925252929313137374141
χ966(751,a) \chi_{ 966 }(751, a) 1-111e(2966)e\left(\frac{29}{66}\right)e(1966)e\left(\frac{19}{66}\right)e(911)e\left(\frac{9}{11}\right)e(4966)e\left(\frac{49}{66}\right)e(1766)e\left(\frac{17}{66}\right)e(2933)e\left(\frac{29}{33}\right)e(1011)e\left(\frac{10}{11}\right)e(3233)e\left(\frac{32}{33}\right)e(5966)e\left(\frac{59}{66}\right)e(311)e\left(\frac{3}{11}\right)
Copy content sage:chi.jacobi_sum(n)
 
χ966(751,a)   \chi_{ 966 }(751,a) \; at   a=\;a = e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
τa(χ966(751,))   \tau_{ a }( \chi_{ 966 }(751,·) )\; at   a=\;a = e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
J(χ966(751,),χ966(n,))   J(\chi_{ 966 }(751,·),\chi_{ 966 }(n,·)) \; for   n= \; n = e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
K(a,b,χ966(751,))  K(a,b,\chi_{ 966 }(751,·)) \; at   a,b=\; a,b = e.g. 1,2