↑

Properties

Label 836352.zb
Modulus $836352$
Conductor $278784$
Order $10560$
Real no
Primitive no
Minimal no
Parity odd

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character orbit
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(836352, base_ring=CyclotomicField(10560)) M = H._module chi = DirichletCharacter(H, M([5280,1815,1760,7776])) chi.galois_orbit()
 
Copy content gp:[g,chi] = znchar(Mod(35, 836352)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("836352.35"); order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Basic properties

Modulus: \(836352\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(278784\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(10560\)
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: no, induced from 278784.rb
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: no
Parity: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Related number fields

Field of values: $\Q(\zeta_{10560})$
Copy content comment:Field of values of chi
 
Copy content sage:CyclotomicField(chi.multiplicative_order())
 
Copy content gp:nfinit(polcyclo(charorder(g,chi)))
 
Copy content magma:CyclotomicField(Order(chi));
 
Fixed field: Number field defined by a degree 10560 polynomial (not computed)
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

First 5 of 2560 characters in Galois orbit

Character \(-1\) \(1\) \(5\) \(7\) \(13\) \(17\) \(19\) \(23\) \(25\) \(29\) \(31\) \(35\)
\(\chi_{836352}(35,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5239}{10560}\right)\) \(e\left(\frac{211}{5280}\right)\) \(e\left(\frac{8281}{10560}\right)\) \(e\left(\frac{347}{880}\right)\) \(e\left(\frac{2011}{3520}\right)\) \(e\left(\frac{301}{1056}\right)\) \(e\left(\frac{5239}{5280}\right)\) \(e\left(\frac{8717}{10560}\right)\) \(e\left(\frac{707}{1320}\right)\) \(e\left(\frac{1887}{3520}\right)\)
\(\chi_{836352}(611,\cdot)\) \(-1\) \(1\) \(e\left(\frac{4871}{10560}\right)\) \(e\left(\frac{3779}{5280}\right)\) \(e\left(\frac{2249}{10560}\right)\) \(e\left(\frac{843}{880}\right)\) \(e\left(\frac{1259}{3520}\right)\) \(e\left(\frac{221}{1056}\right)\) \(e\left(\frac{4871}{5280}\right)\) \(e\left(\frac{5053}{10560}\right)\) \(e\left(\frac{163}{1320}\right)\) \(e\left(\frac{623}{3520}\right)\)
\(\chi_{836352}(827,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5909}{10560}\right)\) \(e\left(\frac{4361}{5280}\right)\) \(e\left(\frac{4571}{10560}\right)\) \(e\left(\frac{257}{880}\right)\) \(e\left(\frac{1601}{3520}\right)\) \(e\left(\frac{791}{1056}\right)\) \(e\left(\frac{629}{5280}\right)\) \(e\left(\frac{7927}{10560}\right)\) \(e\left(\frac{937}{1320}\right)\) \(e\left(\frac{1357}{3520}\right)\)
\(\chi_{836352}(899,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1807}{10560}\right)\) \(e\left(\frac{1003}{5280}\right)\) \(e\left(\frac{1153}{10560}\right)\) \(e\left(\frac{611}{880}\right)\) \(e\left(\frac{1043}{3520}\right)\) \(e\left(\frac{565}{1056}\right)\) \(e\left(\frac{1807}{5280}\right)\) \(e\left(\frac{8981}{10560}\right)\) \(e\left(\frac{971}{1320}\right)\) \(e\left(\frac{1271}{3520}\right)\)
\(\chi_{836352}(1547,\cdot)\) \(-1\) \(1\) \(e\left(\frac{8233}{10560}\right)\) \(e\left(\frac{2317}{5280}\right)\) \(e\left(\frac{9607}{10560}\right)\) \(e\left(\frac{549}{880}\right)\) \(e\left(\frac{1797}{3520}\right)\) \(e\left(\frac{883}{1056}\right)\) \(e\left(\frac{2953}{5280}\right)\) \(e\left(\frac{8339}{10560}\right)\) \(e\left(\frac{269}{1320}\right)\) \(e\left(\frac{769}{3520}\right)\)