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Properties

Label 3249.2.a.e
Level $3249$
Weight $2$
Character orbit 3249.a
Self dual yes
Analytic conductor $25.943$
Analytic rank $0$
Dimension $1$
CM discriminant -19
Inner twists $2$

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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 = [1,0,0,-2,1,0,3,0,0,0,5,0,0,0,0,4,7,0,0,-2,0,0,4,0,-4,0,0,-6, 0,0,0,0,0,0,3,0,0,0,0,0,0,0,-1,-10,0,0,-13,0,2,0,0,0,0,0,5,0,0,0,0,0,15, 0,0,-8,0,0,0,-14,0,0,0,0,-11,0,0,0,15,0,0,4,0,0,16,0,7,0,0,0,0,0,0] 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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 361)
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)
 
\(f(q)\) \(=\) \( q - 2 q^{4} + q^{5} + 3 q^{7} + 5 q^{11} + 4 q^{16} + 7 q^{17} - 2 q^{20} + 4 q^{23} - 4 q^{25} - 6 q^{28} + 3 q^{35} - q^{43} - 10 q^{44} - 13 q^{47} + 2 q^{49} + 5 q^{55} + 15 q^{61} - 8 q^{64} - 14 q^{68}+ \cdots - 8 q^{92}+O(q^{100}) \) 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
0
0 0 −2.00000 1.00000 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 \(\Q(\sqrt{-19}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3249.2.a.e 1
3.b odd 2 1 361.2.a.a ✓ 1
12.b even 2 1 5776.2.a.i 1
15.d odd 2 1 9025.2.a.f 1
19.b odd 2 1 CM 3249.2.a.e 1
57.d even 2 1 361.2.a.a ✓ 1
57.f even 6 2 361.2.c.b 2
57.h odd 6 2 361.2.c.b 2
57.j even 18 6 361.2.e.c 6
57.l odd 18 6 361.2.e.c 6
228.b odd 2 1 5776.2.a.i 1
285.b even 2 1 9025.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
361.2.a.a ✓ 1 3.b odd 2 1
361.2.a.a ✓ 1 57.d even 2 1
361.2.c.b 2 57.f even 6 2
361.2.c.b 2 57.h odd 6 2
361.2.e.c 6 57.j even 18 6
361.2.e.c 6 57.l odd 18 6
3249.2.a.e 1 1.a even 1 1 trivial
3249.2.a.e 1 19.b odd 2 1 CM
5776.2.a.i 1 12.b even 2 1
5776.2.a.i 1 228.b odd 2 1
9025.2.a.f 1 15.d odd 2 1
9025.2.a.f 1 285.b even 2 1

Hecke kernels

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

\( T_{2} \) Copy content Toggle raw display
\( T_{5} - 1 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 1 \) Copy content Toggle raw display
$7$ \( T - 3 \) Copy content Toggle raw display
$11$ \( T - 5 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T - 7 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T - 4 \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T + 1 \) Copy content Toggle raw display
$47$ \( T + 13 \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T - 15 \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 11 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T - 16 \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T \) Copy content Toggle raw display
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