Properties

Label 648.2.a.d
Level $648$
Weight $2$
Character orbit 648.a
Self dual yes
Analytic conductor $5.174$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(5.17430605098\)
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: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{5} + O(q^{10}) \) \( q + q^{5} + 4q^{11} - 5q^{13} + 5q^{17} + 8q^{19} + 4q^{23} - 4q^{25} - 3q^{29} - 4q^{31} + 3q^{37} + 6q^{41} + 4q^{43} + 12q^{47} - 7q^{49} + 10q^{53} + 4q^{55} - 8q^{59} - 5q^{61} - 5q^{65} + 8q^{67} - 16q^{71} - 5q^{73} + 4q^{79} - 4q^{83} + 5q^{85} - 3q^{89} + 8q^{95} + 2q^{97} + O(q^{100}) \)

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 1.00000 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

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 648.2.a.d yes 1
3.b odd 2 1 648.2.a.b 1
4.b odd 2 1 1296.2.a.h 1
8.b even 2 1 5184.2.a.k 1
8.d odd 2 1 5184.2.a.l 1
9.c even 3 2 648.2.i.d 2
9.d odd 6 2 648.2.i.f 2
12.b even 2 1 1296.2.a.d 1
24.f even 2 1 5184.2.a.u 1
24.h odd 2 1 5184.2.a.v 1
36.f odd 6 2 1296.2.i.f 2
36.h even 6 2 1296.2.i.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
648.2.a.b 1 3.b odd 2 1
648.2.a.d yes 1 1.a even 1 1 trivial
648.2.i.d 2 9.c even 3 2
648.2.i.f 2 9.d odd 6 2
1296.2.a.d 1 12.b even 2 1
1296.2.a.h 1 4.b odd 2 1
1296.2.i.f 2 36.f odd 6 2
1296.2.i.k 2 36.h even 6 2
5184.2.a.k 1 8.b even 2 1
5184.2.a.l 1 8.d odd 2 1
5184.2.a.u 1 24.f even 2 1
5184.2.a.v 1 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(648))\):

\( T_{5} - 1 \)
\( T_{7} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( -1 + T \)
$7$ \( T \)
$11$ \( -4 + T \)
$13$ \( 5 + T \)
$17$ \( -5 + T \)
$19$ \( -8 + T \)
$23$ \( -4 + T \)
$29$ \( 3 + T \)
$31$ \( 4 + T \)
$37$ \( -3 + T \)
$41$ \( -6 + T \)
$43$ \( -4 + T \)
$47$ \( -12 + T \)
$53$ \( -10 + T \)
$59$ \( 8 + T \)
$61$ \( 5 + T \)
$67$ \( -8 + T \)
$71$ \( 16 + T \)
$73$ \( 5 + T \)
$79$ \( -4 + T \)
$83$ \( 4 + T \)
$89$ \( 3 + T \)
$97$ \( -2 + T \)
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