Properties

Label 4140.3971
Modulus 41404140
Conductor 828828
Order 6666
Real no
Primitive no
Minimal yes
Parity odd

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4140, base_ring=CyclotomicField(66)) M = H._module chi = DirichletCharacter(H, M([33,11,0,51]))
 
Copy content pari:[g,chi] = znchar(Mod(3971,4140))
 

Basic properties

Modulus: 41404140
Conductor: 828828
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 χ828(659,)\chi_{828}(659,\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 4140.df

χ4140(11,)\chi_{4140}(11,\cdot) χ4140(191,)\chi_{4140}(191,\cdot) χ4140(911,)\chi_{4140}(911,\cdot) χ4140(1031,)\chi_{4140}(1031,\cdot) χ4140(1091,)\chi_{4140}(1091,\cdot) χ4140(1211,)\chi_{4140}(1211,\cdot) χ4140(1391,)\chi_{4140}(1391,\cdot) χ4140(1571,)\chi_{4140}(1571,\cdot) χ4140(1631,)\chi_{4140}(1631,\cdot) χ4140(1811,)\chi_{4140}(1811,\cdot) χ4140(2291,)\chi_{4140}(2291,\cdot) χ4140(2351,)\chi_{4140}(2351,\cdot) χ4140(2471,)\chi_{4140}(2471,\cdot) χ4140(2711,)\chi_{4140}(2711,\cdot) χ4140(3011,)\chi_{4140}(3011,\cdot) χ4140(3191,)\chi_{4140}(3191,\cdot) χ4140(3731,)\chi_{4140}(3731,\cdot) χ4140(3791,)\chi_{4140}(3791,\cdot) χ4140(3971,)\chi_{4140}(3971,\cdot) χ4140(4091,)\chi_{4140}(4091,\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

(2071,461,1657,3961)(2071,461,1657,3961)(1,e(16),1,e(1722))(-1,e\left(\frac{1}{6}\right),1,e\left(\frac{17}{22}\right))

First values

aa 1-11177111113131717191929293131373741414343
χ4140(3971,a) \chi_{ 4140 }(3971, a) 1-111e(2833)e\left(\frac{28}{33}\right)e(4166)e\left(\frac{41}{66}\right)e(533)e\left(\frac{5}{33}\right)e(1011)e\left(\frac{10}{11}\right)e(111)e\left(\frac{1}{11}\right)e(566)e\left(\frac{5}{66}\right)e(3166)e\left(\frac{31}{66}\right)e(522)e\left(\frac{5}{22}\right)e(766)e\left(\frac{7}{66}\right)e(133)e\left(\frac{1}{33}\right)
Copy content sage:chi.jacobi_sum(n)
 
χ4140(3971,a)   \chi_{ 4140 }(3971,a) \; at   a=\;a = e.g. 2