Properties

Label 1344.4.a.be
Level $1344$
Weight $4$
Character orbit 1344.a
Self dual yes
Analytic conductor $79.299$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{137}) \)
Defining polynomial: \(x^{2} - x - 34\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 672)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{137}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -3 q^{3} + ( -5 - \beta ) q^{5} + 7 q^{7} + 9 q^{9} +O(q^{10})\) \( q -3 q^{3} + ( -5 - \beta ) q^{5} + 7 q^{7} + 9 q^{9} + ( -11 - \beta ) q^{11} + ( -12 + 2 \beta ) q^{13} + ( 15 + 3 \beta ) q^{15} + ( 15 - \beta ) q^{17} + ( -16 - 4 \beta ) q^{19} -21 q^{21} + ( 41 + 3 \beta ) q^{23} + ( 37 + 10 \beta ) q^{25} -27 q^{27} + ( -18 + 20 \beta ) q^{29} + ( 56 + 8 \beta ) q^{31} + ( 33 + 3 \beta ) q^{33} + ( -35 - 7 \beta ) q^{35} + ( -24 - 14 \beta ) q^{37} + ( 36 - 6 \beta ) q^{39} + ( -35 + \beta ) q^{41} + ( -20 + 20 \beta ) q^{43} + ( -45 - 9 \beta ) q^{45} + ( 210 - 34 \beta ) q^{47} + 49 q^{49} + ( -45 + 3 \beta ) q^{51} + ( -88 + 18 \beta ) q^{53} + ( 192 + 16 \beta ) q^{55} + ( 48 + 12 \beta ) q^{57} + ( -202 - 2 \beta ) q^{59} + ( 78 + 40 \beta ) q^{61} + 63 q^{63} + ( -214 + 2 \beta ) q^{65} + ( 2 + 30 \beta ) q^{67} + ( -123 - 9 \beta ) q^{69} + ( 407 + 5 \beta ) q^{71} + ( -108 + 46 \beta ) q^{73} + ( -111 - 30 \beta ) q^{75} + ( -77 - 7 \beta ) q^{77} + ( 790 - 22 \beta ) q^{79} + 81 q^{81} + ( -232 + 4 \beta ) q^{83} + ( 62 - 10 \beta ) q^{85} + ( 54 - 60 \beta ) q^{87} + ( -579 - 87 \beta ) q^{89} + ( -84 + 14 \beta ) q^{91} + ( -168 - 24 \beta ) q^{93} + ( 628 + 36 \beta ) q^{95} + ( -880 - 62 \beta ) q^{97} + ( -99 - 9 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 6q^{3} - 10q^{5} + 14q^{7} + 18q^{9} + O(q^{10}) \) \( 2q - 6q^{3} - 10q^{5} + 14q^{7} + 18q^{9} - 22q^{11} - 24q^{13} + 30q^{15} + 30q^{17} - 32q^{19} - 42q^{21} + 82q^{23} + 74q^{25} - 54q^{27} - 36q^{29} + 112q^{31} + 66q^{33} - 70q^{35} - 48q^{37} + 72q^{39} - 70q^{41} - 40q^{43} - 90q^{45} + 420q^{47} + 98q^{49} - 90q^{51} - 176q^{53} + 384q^{55} + 96q^{57} - 404q^{59} + 156q^{61} + 126q^{63} - 428q^{65} + 4q^{67} - 246q^{69} + 814q^{71} - 216q^{73} - 222q^{75} - 154q^{77} + 1580q^{79} + 162q^{81} - 464q^{83} + 124q^{85} + 108q^{87} - 1158q^{89} - 168q^{91} - 336q^{93} + 1256q^{95} - 1760q^{97} - 198q^{99} + O(q^{100}) \)

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
6.35235
−5.35235
0 −3.00000 0 −16.7047 0 7.00000 0 9.00000 0
1.2 0 −3.00000 0 6.70470 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.be 2
4.b odd 2 1 1344.4.a.bm 2
8.b even 2 1 672.4.a.m yes 2
8.d odd 2 1 672.4.a.h 2
24.f even 2 1 2016.4.a.i 2
24.h odd 2 1 2016.4.a.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.4.a.h 2 8.d odd 2 1
672.4.a.m yes 2 8.b even 2 1
1344.4.a.be 2 1.a even 1 1 trivial
1344.4.a.bm 2 4.b odd 2 1
2016.4.a.i 2 24.f even 2 1
2016.4.a.j 2 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1344))\):

\( T_{5}^{2} + 10 T_{5} - 112 \)
\( T_{11}^{2} + 22 T_{11} - 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 3 + T )^{2} \)
$5$ \( -112 + 10 T + T^{2} \)
$7$ \( ( -7 + T )^{2} \)
$11$ \( -16 + 22 T + T^{2} \)
$13$ \( -404 + 24 T + T^{2} \)
$17$ \( 88 - 30 T + T^{2} \)
$19$ \( -1936 + 32 T + T^{2} \)
$23$ \( 448 - 82 T + T^{2} \)
$29$ \( -54476 + 36 T + T^{2} \)
$31$ \( -5632 - 112 T + T^{2} \)
$37$ \( -26276 + 48 T + T^{2} \)
$41$ \( 1088 + 70 T + T^{2} \)
$43$ \( -54400 + 40 T + T^{2} \)
$47$ \( -114272 - 420 T + T^{2} \)
$53$ \( -36644 + 176 T + T^{2} \)
$59$ \( 40256 + 404 T + T^{2} \)
$61$ \( -213116 - 156 T + T^{2} \)
$67$ \( -123296 - 4 T + T^{2} \)
$71$ \( 162224 - 814 T + T^{2} \)
$73$ \( -278228 + 216 T + T^{2} \)
$79$ \( 557792 - 1580 T + T^{2} \)
$83$ \( 51632 + 464 T + T^{2} \)
$89$ \( -701712 + 1158 T + T^{2} \)
$97$ \( 247772 + 1760 T + T^{2} \)
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