Properties

Label 1344.4.a.bu
Level $1344$
Weight $4$
Character orbit 1344.a
Self dual yes
Analytic conductor $79.299$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.37341.1
Defining polynomial: \(x^{3} - 57 x - 148\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 q^{3} + ( -2 + \beta_{1} ) q^{5} -7 q^{7} + 9 q^{9} +O(q^{10})\) \( q + 3 q^{3} + ( -2 + \beta_{1} ) q^{5} -7 q^{7} + 9 q^{9} + ( 16 + \beta_{1} - \beta_{2} ) q^{11} + ( -2 + 4 \beta_{1} + \beta_{2} ) q^{13} + ( -6 + 3 \beta_{1} ) q^{15} + ( 18 - \beta_{1} + \beta_{2} ) q^{17} + ( 28 + 2 \beta_{1} + \beta_{2} ) q^{19} -21 q^{21} + ( -16 - 3 \beta_{1} - \beta_{2} ) q^{23} + ( 31 + 4 \beta_{1} + \beta_{2} ) q^{25} + 27 q^{27} + ( -6 + 10 \beta_{1} - 3 \beta_{2} ) q^{29} + ( -24 - 2 \beta_{1} - 3 \beta_{2} ) q^{31} + ( 48 + 3 \beta_{1} - 3 \beta_{2} ) q^{33} + ( 14 - 7 \beta_{1} ) q^{35} + ( -70 + 4 \beta_{1} + \beta_{2} ) q^{37} + ( -6 + 12 \beta_{1} + 3 \beta_{2} ) q^{39} + ( 138 + 5 \beta_{1} - 3 \beta_{2} ) q^{41} + ( 56 - 8 \beta_{1} - 2 \beta_{2} ) q^{43} + ( -18 + 9 \beta_{1} ) q^{45} + ( -24 - 26 \beta_{1} ) q^{47} + 49 q^{49} + ( 54 - 3 \beta_{1} + 3 \beta_{2} ) q^{51} + ( -134 - 12 \beta_{1} - 3 \beta_{2} ) q^{53} + ( 152 + 10 \beta_{1} + 11 \beta_{2} ) q^{55} + ( 84 + 6 \beta_{1} + 3 \beta_{2} ) q^{57} + ( 180 + 38 \beta_{1} ) q^{59} + ( -266 - 30 \beta_{1} + 7 \beta_{2} ) q^{61} -63 q^{63} + ( 580 + 34 \beta_{1} - 6 \beta_{2} ) q^{65} + ( 16 - 38 \beta_{1} + 2 \beta_{2} ) q^{67} + ( -48 - 9 \beta_{1} - 3 \beta_{2} ) q^{69} + ( 152 - 21 \beta_{1} + \beta_{2} ) q^{71} + ( 410 + 18 \beta_{1} + 16 \beta_{2} ) q^{73} + ( 93 + 12 \beta_{1} + 3 \beta_{2} ) q^{75} + ( -112 - 7 \beta_{1} + 7 \beta_{2} ) q^{77} + ( 456 + 26 \beta_{1} - 8 \beta_{2} ) q^{79} + 81 q^{81} + ( 20 + 40 \beta_{1} + 10 \beta_{2} ) q^{83} + ( -220 + 24 \beta_{1} - 11 \beta_{2} ) q^{85} + ( -18 + 30 \beta_{1} - 9 \beta_{2} ) q^{87} + ( 914 - 31 \beta_{1} + 3 \beta_{2} ) q^{89} + ( 14 - 28 \beta_{1} - 7 \beta_{2} ) q^{91} + ( -72 - 6 \beta_{1} - 9 \beta_{2} ) q^{93} + ( 216 + 52 \beta_{1} - 8 \beta_{2} ) q^{95} + ( 650 - 26 \beta_{1} - 12 \beta_{2} ) q^{97} + ( 144 + 9 \beta_{1} - 9 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 9q^{3} - 6q^{5} - 21q^{7} + 27q^{9} + O(q^{10}) \) \( 3q + 9q^{3} - 6q^{5} - 21q^{7} + 27q^{9} + 48q^{11} - 6q^{13} - 18q^{15} + 54q^{17} + 84q^{19} - 63q^{21} - 48q^{23} + 93q^{25} + 81q^{27} - 18q^{29} - 72q^{31} + 144q^{33} + 42q^{35} - 210q^{37} - 18q^{39} + 414q^{41} + 168q^{43} - 54q^{45} - 72q^{47} + 147q^{49} + 162q^{51} - 402q^{53} + 456q^{55} + 252q^{57} + 540q^{59} - 798q^{61} - 189q^{63} + 1740q^{65} + 48q^{67} - 144q^{69} + 456q^{71} + 1230q^{73} + 279q^{75} - 336q^{77} + 1368q^{79} + 243q^{81} + 60q^{83} - 660q^{85} - 54q^{87} + 2742q^{89} + 42q^{91} - 216q^{93} + 648q^{95} + 1950q^{97} + 432q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 57 x - 148\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\( 4 \nu^{2} - 16 \nu - 152 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + 8 \beta_{1} + 152\)\()/4\)

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
−5.47374
−3.13924
8.61298
0 3.00000 0 −12.9475 0 −7.00000 0 9.00000 0
1.2 0 3.00000 0 −8.27848 0 −7.00000 0 9.00000 0
1.3 0 3.00000 0 15.2260 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.bu 3
4.b odd 2 1 1344.4.a.bs 3
8.b even 2 1 672.4.a.p 3
8.d odd 2 1 672.4.a.r yes 3
24.f even 2 1 2016.4.a.v 3
24.h odd 2 1 2016.4.a.u 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.4.a.p 3 8.b even 2 1
672.4.a.r yes 3 8.d odd 2 1
1344.4.a.bs 3 4.b odd 2 1
1344.4.a.bu 3 1.a even 1 1 trivial
2016.4.a.u 3 24.h odd 2 1
2016.4.a.v 3 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1344))\):

\( T_{5}^{3} + 6 T_{5}^{2} - 216 T_{5} - 1632 \)
\( T_{11}^{3} - 48 T_{11}^{2} - 3060 T_{11} + 95488 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( ( -3 + T )^{3} \)
$5$ \( -1632 - 216 T + 6 T^{2} + T^{3} \)
$7$ \( ( 7 + T )^{3} \)
$11$ \( 95488 - 3060 T - 48 T^{2} + T^{3} \)
$13$ \( 63656 - 6756 T + 6 T^{2} + T^{3} \)
$17$ \( 24736 - 2856 T - 54 T^{2} + T^{3} \)
$19$ \( 200256 - 1872 T - 84 T^{2} + T^{3} \)
$23$ \( -187648 - 4500 T + 48 T^{2} + T^{3} \)
$29$ \( 4848088 - 57108 T + 18 T^{2} + T^{3} \)
$31$ \( -2342912 - 30144 T + 72 T^{2} + T^{3} \)
$37$ \( -53576 + 7932 T + 210 T^{2} + T^{3} \)
$41$ \( 4957664 + 18456 T - 414 T^{2} + T^{3} \)
$43$ \( 722944 - 17664 T - 168 T^{2} + T^{3} \)
$47$ \( 17124736 - 152400 T + 72 T^{2} + T^{3} \)
$53$ \( -7840072 - 7044 T + 402 T^{2} + T^{3} \)
$59$ \( -11538688 - 232032 T - 540 T^{2} + T^{3} \)
$61$ \( -170037272 - 184788 T + 798 T^{2} + T^{3} \)
$67$ \( 44046592 - 349776 T - 48 T^{2} + T^{3} \)
$71$ \( 19646112 - 36756 T - 456 T^{2} + T^{3} \)
$73$ \( 640250616 - 438948 T - 1230 T^{2} + T^{3} \)
$79$ \( 182717824 + 225456 T - 1368 T^{2} + T^{3} \)
$83$ \( 90712000 - 675600 T - 60 T^{2} + T^{3} \)
$89$ \( -524337504 + 2246616 T - 2742 T^{2} + T^{3} \)
$97$ \( -50085448 + 638748 T - 1950 T^{2} + T^{3} \)
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