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Properties

Label 2401.1324
Modulus $2401$
Conductor $343$
Order $49$
Real no
Primitive no
Minimal no
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(2401, base_ring=CyclotomicField(98)) M = H._module chi = DirichletCharacter(H, M([2]))
 
Copy content gp:[g,chi] = znchar(Mod(1324, 2401))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2401.1324");
 

Basic properties

Modulus: \(2401\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(343\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(49\)
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 \(\chi_{343}(43,\cdot)\)
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: no
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 2401.i

\(\chi_{2401}(50,\cdot)\) \(\chi_{2401}(99,\cdot)\) \(\chi_{2401}(148,\cdot)\) \(\chi_{2401}(197,\cdot)\) \(\chi_{2401}(246,\cdot)\) \(\chi_{2401}(295,\cdot)\) \(\chi_{2401}(393,\cdot)\) \(\chi_{2401}(442,\cdot)\) \(\chi_{2401}(491,\cdot)\) \(\chi_{2401}(540,\cdot)\) \(\chi_{2401}(589,\cdot)\) \(\chi_{2401}(638,\cdot)\) \(\chi_{2401}(736,\cdot)\) \(\chi_{2401}(785,\cdot)\) \(\chi_{2401}(834,\cdot)\) \(\chi_{2401}(883,\cdot)\) \(\chi_{2401}(932,\cdot)\) \(\chi_{2401}(981,\cdot)\) \(\chi_{2401}(1079,\cdot)\) \(\chi_{2401}(1128,\cdot)\) \(\chi_{2401}(1177,\cdot)\) \(\chi_{2401}(1226,\cdot)\) \(\chi_{2401}(1275,\cdot)\) \(\chi_{2401}(1324,\cdot)\) \(\chi_{2401}(1422,\cdot)\) \(\chi_{2401}(1471,\cdot)\) \(\chi_{2401}(1520,\cdot)\) \(\chi_{2401}(1569,\cdot)\) \(\chi_{2401}(1618,\cdot)\) \(\chi_{2401}(1667,\cdot)\) ...

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: $\Q(\zeta_{49})$
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 49 polynomial
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\(3\) → \(e\left(\frac{1}{49}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 2401 }(1324, a) \) \(1\)\(1\)\(e\left(\frac{47}{49}\right)\)\(e\left(\frac{1}{49}\right)\)\(e\left(\frac{45}{49}\right)\)\(e\left(\frac{29}{49}\right)\)\(e\left(\frac{48}{49}\right)\)\(e\left(\frac{43}{49}\right)\)\(e\left(\frac{2}{49}\right)\)\(e\left(\frac{27}{49}\right)\)\(e\left(\frac{12}{49}\right)\)\(e\left(\frac{46}{49}\right)\)
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)
 
\( \chi_{ 2401 }(1324,a) \;\) at \(\;a = \) e.g. 2