Properties

Label 2156.507
Modulus 21562156
Conductor 196196
Order 4242
Real no
Primitive no
Minimal yes
Parity even

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2156, base_ring=CyclotomicField(42)) M = H._module chi = DirichletCharacter(H, M([21,25,0]))
 
Copy content pari:[g,chi] = znchar(Mod(507,2156))
 

Basic properties

Modulus: 21562156
Conductor: 196196
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: 4242
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from χ196(115,)\chi_{196}(115,\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 2156.bv

χ2156(199,)\chi_{2156}(199,\cdot) χ2156(243,)\chi_{2156}(243,\cdot) χ2156(507,)\chi_{2156}(507,\cdot) χ2156(551,)\chi_{2156}(551,\cdot) χ2156(859,)\chi_{2156}(859,\cdot) χ2156(1123,)\chi_{2156}(1123,\cdot) χ2156(1167,)\chi_{2156}(1167,\cdot) χ2156(1431,)\chi_{2156}(1431,\cdot) χ2156(1475,)\chi_{2156}(1475,\cdot) χ2156(1739,)\chi_{2156}(1739,\cdot) χ2156(2047,)\chi_{2156}(2047,\cdot) χ2156(2091,)\chi_{2156}(2091,\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(ζ21)\Q(\zeta_{21})
Fixed field: Q(ζ196)+\Q(\zeta_{196})^+

Values on generators

(1079,1277,981)(1079,1277,981)(1,e(2542),1)(-1,e\left(\frac{25}{42}\right),1)

First values

aa 1-1113355991313151517171919232325252727
χ2156(507,a) \chi_{ 2156 }(507, a) 1111e(221)e\left(\frac{2}{21}\right)e(1142)e\left(\frac{11}{42}\right)e(421)e\left(\frac{4}{21}\right)e(914)e\left(\frac{9}{14}\right)e(514)e\left(\frac{5}{14}\right)e(3742)e\left(\frac{37}{42}\right)e(13)e\left(\frac{1}{3}\right)e(542)e\left(\frac{5}{42}\right)e(1121)e\left(\frac{11}{21}\right)e(27)e\left(\frac{2}{7}\right)
Copy content sage:chi.jacobi_sum(n)
 
χ2156(507,a)   \chi_{ 2156 }(507,a) \; at   a=\;a = e.g. 2