↑

Properties

Label 152269.400
Modulus $152269$
Conductor $152269$
Order $312$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(152269, base_ring=CyclotomicField(312)) M = H._module chi = DirichletCharacter(H, M([44,39,276]))
 
Copy content gp:[g,chi] = znchar(Mod(400, 152269))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("152269.400");
 

Basic properties

Modulus: \(152269\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(152269\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(312\)
Copy content comment:Order
 
Copy content sage:chi.multiplicative_order()
 
Copy content gp:charorder(g,chi)
 
Copy content magma:Order(chi);
 
Real: no
Copy content comment:Whether the character is real
 
Copy content sage:chi.multiplicative_order() <= 2
 
Copy content gp:charorder(g,chi) <= 2
 
Copy content magma:Order(chi) le 2;
 
Primitive: yes
Copy content comment:If the character is primitive
 
Copy content sage:chi.is_primitive()
 
Copy content gp:#znconreyconductor(g,chi)==1
 
Copy content magma:IsPrimitive(chi);
 
Minimal: yes
Parity: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 152269.cmn

\(\chi_{152269}(400,\cdot)\) \(\chi_{152269}(621,\cdot)\) \(\chi_{152269}(1226,\cdot)\) \(\chi_{152269}(1596,\cdot)\) \(\chi_{152269}(1895,\cdot)\) \(\chi_{152269}(3878,\cdot)\) \(\chi_{152269}(5516,\cdot)\) \(\chi_{152269}(8876,\cdot)\) \(\chi_{152269}(9630,\cdot)\) \(\chi_{152269}(10293,\cdot)\) \(\chi_{152269}(10657,\cdot)\) \(\chi_{152269}(12068,\cdot)\) \(\chi_{152269}(14486,\cdot)\) \(\chi_{152269}(14635,\cdot)\) \(\chi_{152269}(15461,\cdot)\) \(\chi_{152269}(16182,\cdot)\) \(\chi_{152269}(18685,\cdot)\) \(\chi_{152269}(19140,\cdot)\) \(\chi_{152269}(21259,\cdot)\) \(\chi_{152269}(21792,\cdot)\) \(\chi_{152269}(26790,\cdot)\) \(\chi_{152269}(28467,\cdot)\) \(\chi_{152269}(28766,\cdot)\) \(\chi_{152269}(31067,\cdot)\) \(\chi_{152269}(32387,\cdot)\) \(\chi_{152269}(32400,\cdot)\) \(\chi_{152269}(36501,\cdot)\) \(\chi_{152269}(36599,\cdot)\) \(\chi_{152269}(37164,\cdot)\) \(\chi_{152269}(37528,\cdot)\) ...

Copy content comment:Galois orbit
 
Copy content sage:chi.galois_orbit()
 
Copy content gp:order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 
Copy content magma:order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Related number fields

Field of values: $\Q(\zeta_{312})$
Fixed field: Number field defined by a degree 312 polynomial (not computed)

Values on generators

\((149567,143313,83318)\) → \((e\left(\frac{11}{78}\right),e\left(\frac{1}{8}\right),e\left(\frac{23}{26}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 152269 }(400, a) \) \(1\)\(1\)\(e\left(\frac{121}{156}\right)\)\(e\left(\frac{203}{312}\right)\)\(e\left(\frac{43}{78}\right)\)\(e\left(\frac{49}{104}\right)\)\(e\left(\frac{133}{312}\right)\)\(e\left(\frac{265}{312}\right)\)\(e\left(\frac{17}{52}\right)\)\(e\left(\frac{47}{156}\right)\)\(e\left(\frac{77}{312}\right)\)\(e\left(\frac{17}{24}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x)
 
Copy content gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
 
Copy content magma:chi(x)
 
\( \chi_{ 152269 }(400,a) \;\) at \(\;a = \) e.g. 2