Properties

Label 950.61
Modulus 950950
Conductor 475475
Order 4545
Real no
Primitive no
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(950, base_ring=CyclotomicField(90)) M = H._module chi = DirichletCharacter(H, M([72,10]))
 
Copy content pari:[g,chi] = znchar(Mod(61,950))
 

Basic properties

Modulus: 950950
Conductor: 475475
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: 4545
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from χ475(61,)\chi_{475}(61,\cdot)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 950.bc

χ950(61,)\chi_{950}(61,\cdot) χ950(81,)\chi_{950}(81,\cdot) χ950(111,)\chi_{950}(111,\cdot) χ950(131,)\chi_{950}(131,\cdot) χ950(161,)\chi_{950}(161,\cdot) χ950(271,)\chi_{950}(271,\cdot) χ950(291,)\chi_{950}(291,\cdot) χ950(321,)\chi_{950}(321,\cdot) χ950(441,)\chi_{950}(441,\cdot) χ950(461,)\chi_{950}(461,\cdot) χ950(481,)\chi_{950}(481,\cdot) χ950(491,)\chi_{950}(491,\cdot) χ950(511,)\chi_{950}(511,\cdot) χ950(541,)\chi_{950}(541,\cdot) χ950(631,)\chi_{950}(631,\cdot) χ950(671,)\chi_{950}(671,\cdot) χ950(681,)\chi_{950}(681,\cdot) χ950(731,)\chi_{950}(731,\cdot) χ950(821,)\chi_{950}(821,\cdot) χ950(841,)\chi_{950}(841,\cdot) χ950(861,)\chi_{950}(861,\cdot) χ950(871,)\chi_{950}(871,\cdot) χ950(891,)\chi_{950}(891,\cdot) χ950(921,)\chi_{950}(921,\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(ζ45)\Q(\zeta_{45})
Fixed field: Number field defined by a degree 45 polynomial

Values on generators

(77,401)(77,401)(e(45),e(19))(e\left(\frac{4}{5}\right),e\left(\frac{1}{9}\right))

First values

aa 1-1113377991111131317172121232327272929
χ950(61,a) \chi_{ 950 }(61, a) 1111e(245)e\left(\frac{2}{45}\right)e(23)e\left(\frac{2}{3}\right)e(445)e\left(\frac{4}{45}\right)e(215)e\left(\frac{2}{15}\right)e(3445)e\left(\frac{34}{45}\right)e(2345)e\left(\frac{23}{45}\right)e(3245)e\left(\frac{32}{45}\right)e(145)e\left(\frac{1}{45}\right)e(215)e\left(\frac{2}{15}\right)e(2245)e\left(\frac{22}{45}\right)
Copy content sage:chi.jacobi_sum(n)
 
χ950(61,a)   \chi_{ 950 }(61,a) \; at   a=\;a = e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
τa(χ950(61,))   \tau_{ a }( \chi_{ 950 }(61,·) )\; at   a=\;a = e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
J(χ950(61,),χ950(n,))   J(\chi_{ 950 }(61,·),\chi_{ 950 }(n,·)) \; for   n= \; n = e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
K(a,b,χ950(61,))  K(a,b,\chi_{ 950 }(61,·)) \; at   a,b=\; a,b = e.g. 1,2