Properties

Label 1171.307
Modulus 11711171
Conductor 11711171
Order 234234
Real no
Primitive yes
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(1171, base_ring=CyclotomicField(234)) M = H._module chi = DirichletCharacter(H, M([211]))
 
Copy content pari:[g,chi] = znchar(Mod(307,1171))
 

Basic properties

Modulus: 11711171
Conductor: 11711171
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: 234234
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 1171.u

χ1171(3,)\chi_{1171}(3,\cdot) χ1171(7,)\chi_{1171}(7,\cdot) χ1171(29,)\chi_{1171}(29,\cdot) χ1171(32,)\chi_{1171}(32,\cdot) χ1171(60,)\chi_{1171}(60,\cdot) χ1171(63,)\chi_{1171}(63,\cdot) χ1171(89,)\chi_{1171}(89,\cdot) χ1171(130,)\chi_{1171}(130,\cdot) χ1171(140,)\chi_{1171}(140,\cdot) χ1171(142,)\chi_{1171}(142,\cdot) χ1171(147,)\chi_{1171}(147,\cdot) χ1171(152,)\chi_{1171}(152,\cdot) χ1171(176,)\chi_{1171}(176,\cdot) χ1171(183,)\chi_{1171}(183,\cdot) χ1171(243,)\chi_{1171}(243,\cdot) χ1171(248,)\chi_{1171}(248,\cdot) χ1171(250,)\chi_{1171}(250,\cdot) χ1171(258,)\chi_{1171}(258,\cdot) χ1171(276,)\chi_{1171}(276,\cdot) χ1171(285,)\chi_{1171}(285,\cdot) χ1171(307,)\chi_{1171}(307,\cdot) χ1171(316,)\chi_{1171}(316,\cdot) χ1171(326,)\chi_{1171}(326,\cdot) χ1171(330,)\chi_{1171}(330,\cdot) χ1171(397,)\chi_{1171}(397,\cdot) χ1171(410,)\chi_{1171}(410,\cdot) χ1171(413,)\chi_{1171}(413,\cdot) χ1171(419,)\chi_{1171}(419,\cdot) χ1171(422,)\chi_{1171}(422,\cdot) χ1171(458,)\chi_{1171}(458,\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(ζ117)\Q(\zeta_{117})
Fixed field: Number field defined by a degree 234 polynomial (not computed)

Values on generators

22e(211234)e\left(\frac{211}{234}\right)

First values

aa 1-111223344556677889910101111
χ1171(307,a) \chi_{ 1171 }(307, a) 1-111e(211234)e\left(\frac{211}{234}\right)e(179234)e\left(\frac{179}{234}\right)e(94117)e\left(\frac{94}{117}\right)e(29)e\left(\frac{2}{9}\right)e(23)e\left(\frac{2}{3}\right)e(205234)e\left(\frac{205}{234}\right)e(5578)e\left(\frac{55}{78}\right)e(62117)e\left(\frac{62}{117}\right)e(29234)e\left(\frac{29}{234}\right)e(1978)e\left(\frac{19}{78}\right)
Copy content sage:chi.jacobi_sum(n)
 
χ1171(307,a)   \chi_{ 1171 }(307,a) \; at   a=\;a = e.g. 2