Properties

Label 1344.4.a.bh
Level $1344$
Weight $4$
Character orbit 1344.a
Self dual yes
Analytic conductor $79.299$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{37}) \)
Defining polynomial: \(x^{2} - x - 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
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 \(\beta = 2\sqrt{37}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -3 q^{3} + ( 2 + \beta ) q^{5} + 7 q^{7} + 9 q^{9} +O(q^{10})\) \( q -3 q^{3} + ( 2 + \beta ) q^{5} + 7 q^{7} + 9 q^{9} + ( 24 - \beta ) q^{11} + ( 26 - 4 \beta ) q^{13} + ( -6 - 3 \beta ) q^{15} + ( -6 + 7 \beta ) q^{17} + ( 80 - 2 \beta ) q^{19} -21 q^{21} + ( -24 - 5 \beta ) q^{23} + ( 27 + 4 \beta ) q^{25} -27 q^{27} + ( 190 - 6 \beta ) q^{29} + ( -92 + 18 \beta ) q^{31} + ( -72 + 3 \beta ) q^{33} + ( 14 + 7 \beta ) q^{35} + ( -42 + 20 \beta ) q^{37} + ( -78 + 12 \beta ) q^{39} + ( 26 - 3 \beta ) q^{41} + ( -192 + 16 \beta ) q^{43} + ( 18 + 9 \beta ) q^{45} + ( -32 - 38 \beta ) q^{47} + 49 q^{49} + ( 18 - 21 \beta ) q^{51} + ( 326 + 36 \beta ) q^{53} + ( -100 + 22 \beta ) q^{55} + ( -240 + 6 \beta ) q^{57} + ( 212 - 22 \beta ) q^{59} + ( 530 - 6 \beta ) q^{61} + 63 q^{63} + ( -540 + 18 \beta ) q^{65} + ( -656 + 30 \beta ) q^{67} + ( 72 + 15 \beta ) q^{69} + ( -336 - 51 \beta ) q^{71} + ( -542 + 2 \beta ) q^{73} + ( -81 - 12 \beta ) q^{75} + ( 168 - 7 \beta ) q^{77} + ( 368 + 38 \beta ) q^{79} + 81 q^{81} + ( -140 + 40 \beta ) q^{83} + ( 1024 + 8 \beta ) q^{85} + ( -570 + 18 \beta ) q^{87} + ( 474 - 87 \beta ) q^{89} + ( 182 - 28 \beta ) q^{91} + ( 276 - 54 \beta ) q^{93} + ( -136 + 76 \beta ) q^{95} + ( 402 - 26 \beta ) q^{97} + ( 216 - 9 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 6q^{3} + 4q^{5} + 14q^{7} + 18q^{9} + O(q^{10}) \) \( 2q - 6q^{3} + 4q^{5} + 14q^{7} + 18q^{9} + 48q^{11} + 52q^{13} - 12q^{15} - 12q^{17} + 160q^{19} - 42q^{21} - 48q^{23} + 54q^{25} - 54q^{27} + 380q^{29} - 184q^{31} - 144q^{33} + 28q^{35} - 84q^{37} - 156q^{39} + 52q^{41} - 384q^{43} + 36q^{45} - 64q^{47} + 98q^{49} + 36q^{51} + 652q^{53} - 200q^{55} - 480q^{57} + 424q^{59} + 1060q^{61} + 126q^{63} - 1080q^{65} - 1312q^{67} + 144q^{69} - 672q^{71} - 1084q^{73} - 162q^{75} + 336q^{77} + 736q^{79} + 162q^{81} - 280q^{83} + 2048q^{85} - 1140q^{87} + 948q^{89} + 364q^{91} + 552q^{93} - 272q^{95} + 804q^{97} + 432q^{99} + O(q^{100}) \)

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
−2.54138
3.54138
0 −3.00000 0 −10.1655 0 7.00000 0 9.00000 0
1.2 0 −3.00000 0 14.1655 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.bh 2
4.b odd 2 1 1344.4.a.bp 2
8.b even 2 1 672.4.a.k yes 2
8.d odd 2 1 672.4.a.f 2
24.f even 2 1 2016.4.a.m 2
24.h odd 2 1 2016.4.a.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.4.a.f 2 8.d odd 2 1
672.4.a.k yes 2 8.b even 2 1
1344.4.a.bh 2 1.a even 1 1 trivial
1344.4.a.bp 2 4.b odd 2 1
2016.4.a.m 2 24.f even 2 1
2016.4.a.n 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1344))\):

\( T_{5}^{2} - 4 T_{5} - 144 \)
\( T_{11}^{2} - 48 T_{11} + 428 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 3 + T )^{2} \)
$5$ \( -144 - 4 T + T^{2} \)
$7$ \( ( -7 + T )^{2} \)
$11$ \( 428 - 48 T + T^{2} \)
$13$ \( -1692 - 52 T + T^{2} \)
$17$ \( -7216 + 12 T + T^{2} \)
$19$ \( 5808 - 160 T + T^{2} \)
$23$ \( -3124 + 48 T + T^{2} \)
$29$ \( 30772 - 380 T + T^{2} \)
$31$ \( -39488 + 184 T + T^{2} \)
$37$ \( -57436 + 84 T + T^{2} \)
$41$ \( -656 - 52 T + T^{2} \)
$43$ \( -1024 + 384 T + T^{2} \)
$47$ \( -212688 + 64 T + T^{2} \)
$53$ \( -85532 - 652 T + T^{2} \)
$59$ \( -26688 - 424 T + T^{2} \)
$61$ \( 275572 - 1060 T + T^{2} \)
$67$ \( 297136 + 1312 T + T^{2} \)
$71$ \( -272052 + 672 T + T^{2} \)
$73$ \( 293172 + 1084 T + T^{2} \)
$79$ \( -78288 - 736 T + T^{2} \)
$83$ \( -217200 + 280 T + T^{2} \)
$89$ \( -895536 - 948 T + T^{2} \)
$97$ \( 61556 - 804 T + T^{2} \)
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