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Properties

Label 3249.2.a.l
Level $3249$
Weight $2$
Character orbit 3249.a
Self dual yes
Analytic conductor $25.943$
Analytic rank $1$
Dimension $2$
CM discriminant -19
Inner twists $4$

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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 = [2,0,0,-4,0,0,-6,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,28,0,0,12, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,-2,0,0,0,0,0,4,0,0,0,0,0,-38,0,0,0,0,0,-30, 0,0,-16,0,0,0,0,0,0,0,0,-22,0,0,0,0,0,0,0,0,0,0,0,38,0,0,0,0,0,0] 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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{19}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 19 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-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 \(\beta = \sqrt{19}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{4} + \beta q^{5} - 3 q^{7} - \beta q^{11} + 4 q^{16} + \beta q^{17} - 2 \beta q^{20} - 2 \beta q^{23} + 14 q^{25} + 6 q^{28} - 3 \beta q^{35} - q^{43} + 2 \beta q^{44} - \beta q^{47} + 2 q^{49} + \cdots + 4 \beta q^{92} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4} - 6 q^{7} + 8 q^{16} + 28 q^{25} + 12 q^{28} - 2 q^{43} + 4 q^{49} - 38 q^{55} - 30 q^{61} - 16 q^{64} - 22 q^{73} + 38 q^{85}+O(q^{100}) \) 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
−4.35890
4.35890
0 0 −2.00000 −4.35890 0 −3.00000 0 0 0
1.2 0 0 −2.00000 4.35890 0 −3.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)
3.b odd 2 1 inner
57.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3249.2.a.l ✓ 2
3.b odd 2 1 inner 3249.2.a.l ✓ 2
19.b odd 2 1 CM 3249.2.a.l ✓ 2
57.d even 2 1 inner 3249.2.a.l ✓ 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3249.2.a.l ✓ 2 1.a even 1 1 trivial
3249.2.a.l ✓ 2 3.b odd 2 1 inner
3249.2.a.l ✓ 2 19.b odd 2 1 CM
3249.2.a.l ✓ 2 57.d even 2 1 inner

Hecke kernels

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

\( T_{2} \) Copy content Toggle raw display
\( T_{5}^{2} - 19 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 19 \) Copy content Toggle raw display
$7$ \( (T + 3)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 19 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 19 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 76 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 19 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T + 15)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T + 11)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 76 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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