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Properties

Label 80586.9941
Modulus $80586$
Conductor $40293$
Order $990$
Real no
Primitive no
Minimal yes
Parity even

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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(80586, base_ring=CyclotomicField(990)) M = H._module chi = DirichletCharacter(H, M([825,747,275]))
 
Copy content gp:[g,chi] = znchar(Mod(9941, 80586))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("80586.9941");
 

Basic properties

Modulus: \(80586\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(40293\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(990\)
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 \(\chi_{40293}(9941,\cdot)\)
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 80586.oc

\(\chi_{80586}(95,\cdot)\) \(\chi_{80586}(299,\cdot)\) \(\chi_{80586}(761,\cdot)\) \(\chi_{80586}(1283,\cdot)\) \(\chi_{80586}(1427,\cdot)\) \(\chi_{80586}(1949,\cdot)\) \(\chi_{80586}(2261,\cdot)\) \(\chi_{80586}(2615,\cdot)\) \(\chi_{80586}(2657,\cdot)\) \(\chi_{80586}(3593,\cdot)\) \(\chi_{80586}(3989,\cdot)\) \(\chi_{80586}(4259,\cdot)\) \(\chi_{80586}(4505,\cdot)\) \(\chi_{80586}(4655,\cdot)\) \(\chi_{80586}(4925,\cdot)\) \(\chi_{80586}(5627,\cdot)\) \(\chi_{80586}(5837,\cdot)\) \(\chi_{80586}(6089,\cdot)\) \(\chi_{80586}(6503,\cdot)\) \(\chi_{80586}(6959,\cdot)\) \(\chi_{80586}(7169,\cdot)\) \(\chi_{80586}(7277,\cdot)\) \(\chi_{80586}(7625,\cdot)\) \(\chi_{80586}(8087,\cdot)\) \(\chi_{80586}(8291,\cdot)\) \(\chi_{80586}(8609,\cdot)\) \(\chi_{80586}(8753,\cdot)\) \(\chi_{80586}(9275,\cdot)\) \(\chi_{80586}(9587,\cdot)\) \(\chi_{80586}(9941,\cdot)\) ...

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: $\Q(\zeta_{495})$
Fixed field: Number field defined by a degree 990 polynomial (not computed)

Values on generators

\((71633,1333,47917)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{83}{110}\right),e\left(\frac{5}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 80586 }(9941, a) \) \(1\)\(1\)\(e\left(\frac{194}{495}\right)\)\(e\left(\frac{499}{990}\right)\)\(e\left(\frac{461}{495}\right)\)\(e\left(\frac{413}{990}\right)\)\(e\left(\frac{173}{495}\right)\)\(e\left(\frac{5}{33}\right)\)\(e\left(\frac{388}{495}\right)\)\(e\left(\frac{163}{330}\right)\)\(e\left(\frac{19}{330}\right)\)\(e\left(\frac{887}{990}\right)\)
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)
 
\( \chi_{ 80586 }(9941,a) \;\) at \(\;a = \) e.g. 2