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Properties

Label 9840.8621
Modulus $9840$
Conductor $1968$
Order $40$
Real no
Primitive no
Minimal yes
Parity even

Related objects

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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(9840, base_ring=CyclotomicField(40)) M = H._module chi = DirichletCharacter(H, M([0,30,20,0,3]))
 
Copy content gp:[g,chi] = znchar(Mod(8621, 9840))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9840.8621");
 

Basic properties

Modulus: \(9840\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1968\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(40\)
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 \(\chi_{1968}(749,\cdot)\)
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 9840.pw

\(\chi_{9840}(101,\cdot)\) \(\chi_{9840}(1541,\cdot)\) \(\chi_{9840}(2261,\cdot)\) \(\chi_{9840}(2741,\cdot)\) \(\chi_{9840}(3461,\cdot)\) \(\chi_{9840}(4901,\cdot)\) \(\chi_{9840}(5261,\cdot)\) \(\chi_{9840}(5501,\cdot)\) \(\chi_{9840}(6221,\cdot)\) \(\chi_{9840}(6821,\cdot)\) \(\chi_{9840}(6941,\cdot)\) \(\chi_{9840}(7901,\cdot)\) \(\chi_{9840}(8021,\cdot)\) \(\chi_{9840}(8621,\cdot)\) \(\chi_{9840}(9341,\cdot)\) \(\chi_{9840}(9581,\cdot)\)

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: \(\Q(\zeta_{40})\)
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 40 polynomial
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((1231,7381,3281,3937,1441)\) → \((1,-i,-1,1,e\left(\frac{3}{40}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(43\)
\( \chi_{ 9840 }(8621, a) \) \(1\)\(1\)\(e\left(\frac{17}{40}\right)\)\(e\left(\frac{19}{40}\right)\)\(e\left(\frac{23}{40}\right)\)\(e\left(\frac{39}{40}\right)\)\(e\left(\frac{37}{40}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{11}{40}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{7}{10}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x) # x integer
 
Copy content gp:chareval(g,chi,x) \\ x integer, value in Q/Z
 
Copy content magma:chi(x)
 
\( \chi_{ 9840 }(8621,a) \;\) at \(\;a = \) e.g. 2