Properties

Label 1344.4.a.j
Level $1344$
Weight $4$
Character orbit 1344.a
Self dual yes
Analytic conductor $79.299$
Analytic rank $1$
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: \(1\)
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

\(f(q)\) \(=\) \( q - 3q^{3} + 10q^{5} - 7q^{7} + 9q^{9} + O(q^{10}) \) \( q - 3q^{3} + 10q^{5} - 7q^{7} + 9q^{9} - 12q^{11} - 30q^{13} - 30q^{15} + 34q^{17} + 148q^{19} + 21q^{21} - 152q^{23} - 25q^{25} - 27q^{27} + 106q^{29} - 304q^{31} + 36q^{33} - 70q^{35} + 114q^{37} + 90q^{39} + 202q^{41} + 116q^{43} + 90q^{45} - 224q^{47} + 49q^{49} - 102q^{51} + 274q^{53} - 120q^{55} - 444q^{57} - 660q^{59} - 382q^{61} - 63q^{63} - 300q^{65} + 12q^{67} + 456q^{69} + 552q^{71} - 614q^{73} + 75q^{75} + 84q^{77} - 880q^{79} + 81q^{81} - 108q^{83} + 340q^{85} - 318q^{87} - 86q^{89} + 210q^{91} + 912q^{93} + 1480q^{95} + 1426q^{97} - 108q^{99} + O(q^{100}) \)

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 10.0000 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.j 1
4.b odd 2 1 1344.4.a.y 1
8.b even 2 1 336.4.a.g 1
8.d odd 2 1 168.4.a.a 1
24.f even 2 1 504.4.a.f 1
24.h odd 2 1 1008.4.a.p 1
56.e even 2 1 1176.4.a.m 1
56.h odd 2 1 2352.4.a.n 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.a.a 1 8.d odd 2 1
336.4.a.g 1 8.b even 2 1
504.4.a.f 1 24.f even 2 1
1008.4.a.p 1 24.h odd 2 1
1176.4.a.m 1 56.e even 2 1
1344.4.a.j 1 1.a even 1 1 trivial
1344.4.a.y 1 4.b odd 2 1
2352.4.a.n 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1344))\):

\( T_{5} - 10 \)
\( T_{11} + 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 3 + T \)
$5$ \( -10 + T \)
$7$ \( 7 + T \)
$11$ \( 12 + T \)
$13$ \( 30 + T \)
$17$ \( -34 + T \)
$19$ \( -148 + T \)
$23$ \( 152 + T \)
$29$ \( -106 + T \)
$31$ \( 304 + T \)
$37$ \( -114 + T \)
$41$ \( -202 + T \)
$43$ \( -116 + T \)
$47$ \( 224 + T \)
$53$ \( -274 + T \)
$59$ \( 660 + T \)
$61$ \( 382 + T \)
$67$ \( -12 + T \)
$71$ \( -552 + T \)
$73$ \( 614 + T \)
$79$ \( 880 + T \)
$83$ \( 108 + T \)
$89$ \( 86 + T \)
$97$ \( -1426 + T \)
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