Properties

Label 648.4.a.b
Level $648$
Weight $4$
Character orbit 648.a
Self dual yes
Analytic conductor $38.233$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 648.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(38.2332376837\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 5q^{5} + 36q^{7} + O(q^{10}) \) \( q + 5q^{5} + 36q^{7} - 64q^{11} - 65q^{13} - 59q^{17} - 28q^{19} - 160q^{23} - 100q^{25} + 57q^{29} + 164q^{31} + 180q^{35} - 321q^{37} + 246q^{41} - 8q^{43} - 84q^{47} + 953q^{49} - 478q^{53} - 320q^{55} + 32q^{59} + 415q^{61} - 325q^{65} - 220q^{67} - 884q^{71} - 77q^{73} - 2304q^{77} - 80q^{79} - 1268q^{83} - 295q^{85} - 123q^{89} - 2340q^{91} - 140q^{95} + 1346q^{97} + O(q^{100}) \)

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 5.00000 0 36.0000 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

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 648.4.a.b yes 1
3.b odd 2 1 648.4.a.a 1
4.b odd 2 1 1296.4.a.f 1
9.c even 3 2 648.4.i.c 2
9.d odd 6 2 648.4.i.j 2
12.b even 2 1 1296.4.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
648.4.a.a 1 3.b odd 2 1
648.4.a.b yes 1 1.a even 1 1 trivial
648.4.i.c 2 9.c even 3 2
648.4.i.j 2 9.d odd 6 2
1296.4.a.c 1 12.b even 2 1
1296.4.a.f 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 5 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(648))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( -5 + T \)
$7$ \( -36 + T \)
$11$ \( 64 + T \)
$13$ \( 65 + T \)
$17$ \( 59 + T \)
$19$ \( 28 + T \)
$23$ \( 160 + T \)
$29$ \( -57 + T \)
$31$ \( -164 + T \)
$37$ \( 321 + T \)
$41$ \( -246 + T \)
$43$ \( 8 + T \)
$47$ \( 84 + T \)
$53$ \( 478 + T \)
$59$ \( -32 + T \)
$61$ \( -415 + T \)
$67$ \( 220 + T \)
$71$ \( 884 + T \)
$73$ \( 77 + T \)
$79$ \( 80 + T \)
$83$ \( 1268 + T \)
$89$ \( 123 + T \)
$97$ \( -1346 + T \)
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