Properties

Label 1344.4.a.c
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 672)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 3q^{3} - 6q^{5} - 7q^{7} + 9q^{9} + O(q^{10}) \) \( q - 3q^{3} - 6q^{5} - 7q^{7} + 9q^{9} - 4q^{11} + 46q^{13} + 18q^{15} - 82q^{17} + 84q^{19} + 21q^{21} - 44q^{23} - 89q^{25} - 27q^{27} - 70q^{29} + 152q^{31} + 12q^{33} + 42q^{35} + 146q^{37} - 138q^{39} + 94q^{41} + 488q^{43} - 54q^{45} + 32q^{47} + 49q^{49} + 246q^{51} + 562q^{53} + 24q^{55} - 252q^{57} - 476q^{59} - 34q^{61} - 63q^{63} - 276q^{65} - 520q^{67} + 132q^{69} + 36q^{71} - 654q^{73} + 267q^{75} + 28q^{77} + 608q^{79} + 81q^{81} + 284q^{83} + 492q^{85} + 210q^{87} - 954q^{89} - 322q^{91} - 456q^{93} - 504q^{95} - 1694q^{97} - 36q^{99} + O(q^{100}) \)

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 −6.00000 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.c 1
4.b odd 2 1 1344.4.a.r 1
8.b even 2 1 672.4.a.d yes 1
8.d odd 2 1 672.4.a.b 1
24.f even 2 1 2016.4.a.b 1
24.h odd 2 1 2016.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.4.a.b 1 8.d odd 2 1
672.4.a.d yes 1 8.b even 2 1
1344.4.a.c 1 1.a even 1 1 trivial
1344.4.a.r 1 4.b odd 2 1
2016.4.a.a 1 24.h odd 2 1
2016.4.a.b 1 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1344))\):

\( T_{5} + 6 \)
\( T_{11} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 3 + T \)
$5$ \( 6 + T \)
$7$ \( 7 + T \)
$11$ \( 4 + T \)
$13$ \( -46 + T \)
$17$ \( 82 + T \)
$19$ \( -84 + T \)
$23$ \( 44 + T \)
$29$ \( 70 + T \)
$31$ \( -152 + T \)
$37$ \( -146 + T \)
$41$ \( -94 + T \)
$43$ \( -488 + T \)
$47$ \( -32 + T \)
$53$ \( -562 + T \)
$59$ \( 476 + T \)
$61$ \( 34 + T \)
$67$ \( 520 + T \)
$71$ \( -36 + T \)
$73$ \( 654 + T \)
$79$ \( -608 + T \)
$83$ \( -284 + T \)
$89$ \( 954 + T \)
$97$ \( 1694 + T \)
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