↑

Properties

Label 30687.2036
Modulus $30687$
Conductor $30687$
Order $832$
Real no
Primitive yes
Minimal yes
Parity odd

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(30687, base_ring=CyclotomicField(832)) M = H._module chi = DirichletCharacter(H, M([416,112,481]))
 
Copy content gp:[g,chi] = znchar(Mod(2036, 30687))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("30687.2036");
 

Basic properties

Modulus: \(30687\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(30687\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(832\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 30687.he

\(\chi_{30687}(20,\cdot)\) \(\chi_{30687}(35,\cdot)\) \(\chi_{30687}(74,\cdot)\) \(\chi_{30687}(104,\cdot)\) \(\chi_{30687}(164,\cdot)\) \(\chi_{30687}(287,\cdot)\) \(\chi_{30687}(299,\cdot)\) \(\chi_{30687}(485,\cdot)\) \(\chi_{30687}(503,\cdot)\) \(\chi_{30687}(614,\cdot)\) \(\chi_{30687}(737,\cdot)\) \(\chi_{30687}(866,\cdot)\) \(\chi_{30687}(1145,\cdot)\) \(\chi_{30687}(1187,\cdot)\) \(\chi_{30687}(1277,\cdot)\) \(\chi_{30687}(1322,\cdot)\) \(\chi_{30687}(1364,\cdot)\) \(\chi_{30687}(1445,\cdot)\) \(\chi_{30687}(1511,\cdot)\) \(\chi_{30687}(1910,\cdot)\) \(\chi_{30687}(1943,\cdot)\) \(\chi_{30687}(2006,\cdot)\) \(\chi_{30687}(2036,\cdot)\) \(\chi_{30687}(2228,\cdot)\) \(\chi_{30687}(2351,\cdot)\) \(\chi_{30687}(2387,\cdot)\) \(\chi_{30687}(2390,\cdot)\) \(\chi_{30687}(2435,\cdot)\) \(\chi_{30687}(2522,\cdot)\) \(\chi_{30687}(2585,\cdot)\) ...

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: $\Q(\zeta_{832})$
Fixed field: Number field defined by a degree 832 polynomial (not computed)

Values on generators

\((20459,7528,10813)\) → \((-1,e\left(\frac{7}{52}\right),e\left(\frac{37}{64}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 30687 }(2036, a) \) \(-1\)\(1\)\(e\left(\frac{121}{416}\right)\)\(e\left(\frac{121}{208}\right)\)\(e\left(\frac{337}{832}\right)\)\(e\left(\frac{1}{104}\right)\)\(e\left(\frac{363}{416}\right)\)\(e\left(\frac{579}{832}\right)\)\(e\left(\frac{87}{832}\right)\)\(e\left(\frac{621}{832}\right)\)\(e\left(\frac{125}{416}\right)\)\(e\left(\frac{17}{104}\right)\)
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)
 
\( \chi_{ 30687 }(2036,a) \;\) at \(\;a = \) e.g. 2