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Properties

Label 987696.59
Modulus $987696$
Conductor $987696$
Order $12996$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(987696, base_ring=CyclotomicField(12996)) M = H._module chi = DirichletCharacter(H, M([6498,3249,10830,362]))
 
Copy content gp:[g,chi] = znchar(Mod(59, 987696))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("987696.59");
 

Basic properties

Modulus: \(987696\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(987696\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(12996\)
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 987696.wd

\(\chi_{987696}(59,\cdot)\) \(\chi_{987696}(371,\cdot)\) \(\chi_{987696}(659,\cdot)\) \(\chi_{987696}(1067,\cdot)\) \(\chi_{987696}(1211,\cdot)\) \(\chi_{987696}(1307,\cdot)\) \(\chi_{987696}(1427,\cdot)\)

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: $\Q(\zeta_{12996})$
Fixed field: Number field defined by a degree 12996 polynomial (not computed)

Values on generators

\((617311,740773,438977,857377)\) → \((-1,i,e\left(\frac{5}{6}\right),e\left(\frac{181}{6498}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 987696 }(59, a) \) \(-1\)\(1\)\(e\left(\frac{3215}{12996}\right)\)\(e\left(\frac{41}{1083}\right)\)\(e\left(\frac{537}{1444}\right)\)\(e\left(\frac{6181}{12996}\right)\)\(e\left(\frac{1747}{6498}\right)\)\(e\left(\frac{149}{6498}\right)\)\(e\left(\frac{3215}{6498}\right)\)\(e\left(\frac{8947}{12996}\right)\)\(e\left(\frac{333}{361}\right)\)\(e\left(\frac{3707}{12996}\right)\)
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)
 
\( \chi_{ 987696 }(59,a) \;\) at \(\;a = \) e.g. 2