Properties

Label 8512.75
Modulus 85128512
Conductor 85128512
Order 4848
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(8512, base_ring=CyclotomicField(48)) M = H._module chi = DirichletCharacter(H, M([24,15,40,24]))
 
Copy content pari:[g,chi] = znchar(Mod(75,8512))
 

Basic properties

Modulus: 85128512
Conductor: 85128512
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: 4848
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 8512.js

χ8512(75,)\chi_{8512}(75,\cdot) χ8512(227,)\chi_{8512}(227,\cdot) χ8512(1139,)\chi_{8512}(1139,\cdot) χ8512(1291,)\chi_{8512}(1291,\cdot) χ8512(2203,)\chi_{8512}(2203,\cdot) χ8512(2355,)\chi_{8512}(2355,\cdot) χ8512(3267,)\chi_{8512}(3267,\cdot) χ8512(3419,)\chi_{8512}(3419,\cdot) χ8512(4331,)\chi_{8512}(4331,\cdot) χ8512(4483,)\chi_{8512}(4483,\cdot) χ8512(5395,)\chi_{8512}(5395,\cdot) χ8512(5547,)\chi_{8512}(5547,\cdot) χ8512(6459,)\chi_{8512}(6459,\cdot) χ8512(6611,)\chi_{8512}(6611,\cdot) χ8512(7523,)\chi_{8512}(7523,\cdot) χ8512(7675,)\chi_{8512}(7675,\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(ζ48)\Q(\zeta_{48})
Fixed field: Number field defined by a degree 48 polynomial

Values on generators

(5055,6917,7297,3137)(5055,6917,7297,3137)(1,e(516),e(56),1)(-1,e\left(\frac{5}{16}\right),e\left(\frac{5}{6}\right),-1)

First values

aa 1-1113355991111131315151717232325252727
χ8512(75,a) \chi_{ 8512 }(75, a) 1-111e(3748)e\left(\frac{37}{48}\right)e(2348)e\left(\frac{23}{48}\right)e(1324)e\left(\frac{13}{24}\right)e(1948)e\left(\frac{19}{48}\right)e(1116)e\left(\frac{11}{16}\right)iie(712)e\left(\frac{7}{12}\right)e(1324)e\left(\frac{13}{24}\right)e(2324)e\left(\frac{23}{24}\right)e(516)e\left(\frac{5}{16}\right)
Copy content sage:chi.jacobi_sum(n)
 
χ8512(75,a)   \chi_{ 8512 }(75,a) \; at   a=\;a = e.g. 2