Properties

Label 288.2.a.e
Level $288$
Weight $2$
Character orbit 288.a
Self dual yes
Analytic conductor $2.300$
Analytic rank $0$
Dimension $1$
CM discriminant -4
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 288.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.29969157821\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

\(f(q)\) \(=\) \( q + 4q^{5} + O(q^{10}) \) \( q + 4q^{5} - 6q^{13} + 8q^{17} + 11q^{25} - 4q^{29} - 2q^{37} - 8q^{41} - 7q^{49} - 4q^{53} - 10q^{61} - 24q^{65} + 6q^{73} + 32q^{85} + 16q^{89} - 18q^{97} + O(q^{100}) \)

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.

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 0 4.00000 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.2.a.e yes 1
3.b odd 2 1 288.2.a.a 1
4.b odd 2 1 CM 288.2.a.e yes 1
5.b even 2 1 7200.2.a.be 1
5.c odd 4 2 7200.2.f.q 2
8.b even 2 1 576.2.a.a 1
8.d odd 2 1 576.2.a.a 1
9.c even 3 2 2592.2.i.a 2
9.d odd 6 2 2592.2.i.x 2
12.b even 2 1 288.2.a.a 1
15.d odd 2 1 7200.2.a.bf 1
15.e even 4 2 7200.2.f.n 2
16.e even 4 2 2304.2.d.l 2
16.f odd 4 2 2304.2.d.l 2
20.d odd 2 1 7200.2.a.be 1
20.e even 4 2 7200.2.f.q 2
24.f even 2 1 576.2.a.i 1
24.h odd 2 1 576.2.a.i 1
36.f odd 6 2 2592.2.i.a 2
36.h even 6 2 2592.2.i.x 2
48.i odd 4 2 2304.2.d.h 2
48.k even 4 2 2304.2.d.h 2
60.h even 2 1 7200.2.a.bf 1
60.l odd 4 2 7200.2.f.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.a.a 1 3.b odd 2 1
288.2.a.a 1 12.b even 2 1
288.2.a.e yes 1 1.a even 1 1 trivial
288.2.a.e yes 1 4.b odd 2 1 CM
576.2.a.a 1 8.b even 2 1
576.2.a.a 1 8.d odd 2 1
576.2.a.i 1 24.f even 2 1
576.2.a.i 1 24.h odd 2 1
2304.2.d.h 2 48.i odd 4 2
2304.2.d.h 2 48.k even 4 2
2304.2.d.l 2 16.e even 4 2
2304.2.d.l 2 16.f odd 4 2
2592.2.i.a 2 9.c even 3 2
2592.2.i.a 2 36.f odd 6 2
2592.2.i.x 2 9.d odd 6 2
2592.2.i.x 2 36.h even 6 2
7200.2.a.be 1 5.b even 2 1
7200.2.a.be 1 20.d odd 2 1
7200.2.a.bf 1 15.d odd 2 1
7200.2.a.bf 1 60.h even 2 1
7200.2.f.n 2 15.e even 4 2
7200.2.f.n 2 60.l odd 4 2
7200.2.f.q 2 5.c odd 4 2
7200.2.f.q 2 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(288))\):

\( T_{5} - 4 \)
\( T_{7} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( -4 + T \)
$7$ \( T \)
$11$ \( T \)
$13$ \( 6 + T \)
$17$ \( -8 + T \)
$19$ \( T \)
$23$ \( T \)
$29$ \( 4 + T \)
$31$ \( T \)
$37$ \( 2 + T \)
$41$ \( 8 + T \)
$43$ \( T \)
$47$ \( T \)
$53$ \( 4 + T \)
$59$ \( T \)
$61$ \( 10 + T \)
$67$ \( T \)
$71$ \( T \)
$73$ \( -6 + T \)
$79$ \( T \)
$83$ \( T \)
$89$ \( -16 + T \)
$97$ \( 18 + T \)
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