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Properties

Label 3249.2.a.k
Level $3249$
Weight $2$
Character orbit 3249.a
Self dual yes
Analytic conductor $25.943$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3249,2,Mod(1,3249)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3249.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3249, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3249 = 3^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3249.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-1,0,5,3,0,-1,-9,0,-10,-5,0,8,-8,0,3,-3,0,0,16,0,-6,6,0,3, -4,0,6,-2,0,4,-9,0,10,7,0,4,0,0,-22,10,0,3,-4,0,14,9,0,-5,-27,0,20,-6, 0,1,-4,0,18,24,0,9,32,0,7,12,0,16,-16,0,-12,-4,0,15,32,0,0,11,0,4,30,0, -22,6,0,-13,24,0,14,-6,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(91)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(25.9433956167\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1083)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} + (\beta + 2) q^{4} + (\beta + 1) q^{5} + (\beta - 1) q^{7} + ( - \beta - 4) q^{8} + ( - 2 \beta - 4) q^{10} + (\beta - 3) q^{11} + 4 q^{13} - 4 q^{14} + 3 \beta q^{16} + ( - \beta - 1) q^{17} + \cdots + (3 \beta + 4) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 5 q^{4} + 3 q^{5} - q^{7} - 9 q^{8} - 10 q^{10} - 5 q^{11} + 8 q^{13} - 8 q^{14} + 3 q^{16} - 3 q^{17} + 16 q^{20} - 6 q^{22} + 6 q^{23} + 3 q^{25} - 4 q^{26} + 6 q^{28} - 2 q^{29} + 4 q^{31}+ \cdots + 11 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.56155
−1.56155
−2.56155 0 4.56155 3.56155 0 1.56155 −6.56155 0 −9.12311
1.2 1.56155 0 0.438447 −0.561553 0 −2.56155 −2.43845 0 −0.876894
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(19\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3249.2.a.k 2
3.b odd 2 1 1083.2.a.j yes 2
19.b odd 2 1 3249.2.a.q 2
57.d even 2 1 1083.2.a.g ✓ 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1083.2.a.g ✓ 2 57.d even 2 1
1083.2.a.j yes 2 3.b odd 2 1
3249.2.a.k 2 1.a even 1 1 trivial
3249.2.a.q 2 19.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3249))\):

\( T_{2}^{2} + T_{2} - 4 \) Copy content Toggle raw display
\( T_{5}^{2} - 3T_{5} - 2 \) Copy content Toggle raw display
\( T_{13} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T - 4 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 3T - 2 \) Copy content Toggle raw display
$7$ \( T^{2} + T - 4 \) Copy content Toggle raw display
$11$ \( T^{2} + 5T + 2 \) Copy content Toggle raw display
$13$ \( (T - 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 3T - 2 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 6T - 8 \) Copy content Toggle raw display
$29$ \( T^{2} + 2T - 16 \) Copy content Toggle raw display
$31$ \( T^{2} - 4T - 64 \) Copy content Toggle raw display
$37$ \( T^{2} - 4T - 64 \) Copy content Toggle raw display
$41$ \( T^{2} - 10T + 8 \) Copy content Toggle raw display
$43$ \( T^{2} - 3T - 36 \) Copy content Toggle raw display
$47$ \( T^{2} - 9T - 18 \) Copy content Toggle raw display
$53$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
$59$ \( (T - 12)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 9T - 18 \) Copy content Toggle raw display
$67$ \( (T - 8)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 4T - 64 \) Copy content Toggle raw display
$73$ \( T^{2} - 15T + 18 \) Copy content Toggle raw display
$79$ \( T^{2} - 4T - 64 \) Copy content Toggle raw display
$83$ \( T^{2} - 6T - 144 \) Copy content Toggle raw display
$89$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
$97$ \( (T - 4)^{2} \) Copy content Toggle raw display
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