Properties

Label 1344.4.a.z
Level $1344$
Weight $4$
Character orbit 1344.a
Self dual yes
Analytic conductor $79.299$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1344.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(79.2985670477\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 3q^{3} + 16q^{5} - 7q^{7} + 9q^{9} + O(q^{10}) \) \( q + 3q^{3} + 16q^{5} - 7q^{7} + 9q^{9} - 18q^{11} + 54q^{13} + 48q^{15} - 128q^{17} + 52q^{19} - 21q^{21} + 202q^{23} + 131q^{25} + 27q^{27} - 302q^{29} + 200q^{31} - 54q^{33} - 112q^{35} + 150q^{37} + 162q^{39} + 172q^{41} + 164q^{43} + 144q^{45} + 460q^{47} + 49q^{49} - 384q^{51} + 190q^{53} - 288q^{55} + 156q^{57} + 96q^{59} - 622q^{61} - 63q^{63} + 864q^{65} + 744q^{67} + 606q^{69} + 54q^{71} + 742q^{73} + 393q^{75} + 126q^{77} + 92q^{79} + 81q^{81} - 228q^{83} - 2048q^{85} - 906q^{87} - 116q^{89} - 378q^{91} + 600q^{93} + 832q^{95} - 554q^{97} - 162q^{99} + O(q^{100}) \)

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 3.00000 0 16.0000 0 −7.00000 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(1\)

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 1344.4.a.z 1
4.b odd 2 1 1344.4.a.l 1
8.b even 2 1 336.4.a.a 1
8.d odd 2 1 168.4.a.d 1
24.f even 2 1 504.4.a.h 1
24.h odd 2 1 1008.4.a.t 1
56.e even 2 1 1176.4.a.h 1
56.h odd 2 1 2352.4.a.bj 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.a.d 1 8.d odd 2 1
336.4.a.a 1 8.b even 2 1
504.4.a.h 1 24.f even 2 1
1008.4.a.t 1 24.h odd 2 1
1176.4.a.h 1 56.e even 2 1
1344.4.a.l 1 4.b odd 2 1
1344.4.a.z 1 1.a even 1 1 trivial
2352.4.a.bj 1 56.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1344))\):

\( T_{5} - 16 \)
\( T_{11} + 18 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -3 + T \)
$5$ \( -16 + T \)
$7$ \( 7 + T \)
$11$ \( 18 + T \)
$13$ \( -54 + T \)
$17$ \( 128 + T \)
$19$ \( -52 + T \)
$23$ \( -202 + T \)
$29$ \( 302 + T \)
$31$ \( -200 + T \)
$37$ \( -150 + T \)
$41$ \( -172 + T \)
$43$ \( -164 + T \)
$47$ \( -460 + T \)
$53$ \( -190 + T \)
$59$ \( -96 + T \)
$61$ \( 622 + T \)
$67$ \( -744 + T \)
$71$ \( -54 + T \)
$73$ \( -742 + T \)
$79$ \( -92 + T \)
$83$ \( 228 + T \)
$89$ \( 116 + T \)
$97$ \( 554 + T \)
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