Properties

Label 1197.701
Modulus 11971197
Conductor 5757
Order 1818
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(1197, base_ring=CyclotomicField(18)) M = H._module chi = DirichletCharacter(H, M([9,0,10]))
 
Copy content pari:[g,chi] = znchar(Mod(701,1197))
 

Basic properties

Modulus: 11971197
Conductor: 5757
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: 1818
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from χ57(17,)\chi_{57}(17,\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 1197.gk

χ1197(386,)\chi_{1197}(386,\cdot) χ1197(575,)\chi_{1197}(575,\cdot) χ1197(701,)\chi_{1197}(701,\cdot) χ1197(764,)\chi_{1197}(764,\cdot) χ1197(890,)\chi_{1197}(890,\cdot) χ1197(1016,)\chi_{1197}(1016,\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(ζ9)\Q(\zeta_{9})
Fixed field: 18.0.5677392343251487443465123.1

Values on generators

(533,514,1009)(533,514,1009)(1,1,e(59))(-1,1,e\left(\frac{5}{9}\right))

First values

aa 1-11122445588101011111313161617172020
χ1197(701,a) \chi_{ 1197 }(701, a) 1-111e(118)e\left(\frac{1}{18}\right)e(19)e\left(\frac{1}{9}\right)e(718)e\left(\frac{7}{18}\right)e(16)e\left(\frac{1}{6}\right)e(49)e\left(\frac{4}{9}\right)e(16)e\left(\frac{1}{6}\right)e(79)e\left(\frac{7}{9}\right)e(29)e\left(\frac{2}{9}\right)e(118)e\left(\frac{1}{18}\right)1-1
Copy content sage:chi.jacobi_sum(n)
 
χ1197(701,a)   \chi_{ 1197 }(701,a) \; at   a=\;a = e.g. 2