Properties

Label 1344.4.a.bv
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: 3.3.22700.1
Defining polynomial: \(x^{3} - x^{2} - 28 x + 12\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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.

\(f(q)\) \(=\) \( q + 3 q^{3} + ( 3 + \beta_{2} ) q^{5} + 7 q^{7} + 9 q^{9} +O(q^{10})\) \( q + 3 q^{3} + ( 3 + \beta_{2} ) q^{5} + 7 q^{7} + 9 q^{9} + ( 1 - \beta_{1} - 2 \beta_{2} ) q^{11} + ( 16 + \beta_{1} + \beta_{2} ) q^{13} + ( 9 + 3 \beta_{2} ) q^{15} + ( 9 - \beta_{1} + 4 \beta_{2} ) q^{17} + ( 48 + \beta_{1} - 5 \beta_{2} ) q^{19} + 21 q^{21} + ( -17 - 5 \beta_{1} ) q^{23} + ( 105 + 5 \beta_{1} + 5 \beta_{2} ) q^{25} + 27 q^{27} + ( -98 - 5 \beta_{1} + \beta_{2} ) q^{29} + ( 24 + 7 \beta_{1} + \beta_{2} ) q^{31} + ( 3 - 3 \beta_{1} - 6 \beta_{2} ) q^{33} + ( 21 + 7 \beta_{2} ) q^{35} + ( -8 + 3 \beta_{1} + 11 \beta_{2} ) q^{37} + ( 48 + 3 \beta_{1} + 3 \beta_{2} ) q^{39} + ( 127 - 5 \beta_{1} + 14 \beta_{2} ) q^{41} + ( 264 - 8 \beta_{2} ) q^{43} + ( 27 + 9 \beta_{2} ) q^{45} + ( -82 - 2 \beta_{2} ) q^{47} + 49 q^{49} + ( 27 - 3 \beta_{1} + 12 \beta_{2} ) q^{51} + ( 11 \beta_{1} - 21 \beta_{2} ) q^{53} + ( -440 - 15 \beta_{1} - 25 \beta_{2} ) q^{55} + ( 144 + 3 \beta_{1} - 15 \beta_{2} ) q^{57} + ( 494 + 4 \beta_{1} + 14 \beta_{2} ) q^{59} + ( 258 - 5 \beta_{1} + 41 \beta_{2} ) q^{61} + 63 q^{63} + ( 270 + 10 \beta_{1} + 40 \beta_{2} ) q^{65} + ( 422 + 6 \beta_{2} ) q^{67} + ( -51 - 15 \beta_{1} ) q^{69} + ( -63 + 25 \beta_{1} + 4 \beta_{2} ) q^{71} + ( -384 - 20 \beta_{1} + 2 \beta_{2} ) q^{73} + ( 315 + 15 \beta_{1} + 15 \beta_{2} ) q^{75} + ( 7 - 7 \beta_{1} - 14 \beta_{2} ) q^{77} + ( -278 + 8 \beta_{1} - 14 \beta_{2} ) q^{79} + 81 q^{81} + ( 520 - 10 \beta_{1} + 14 \beta_{2} ) q^{83} + ( 910 + 15 \beta_{1} - 5 \beta_{2} ) q^{85} + ( -294 - 15 \beta_{1} + 3 \beta_{2} ) q^{87} + ( -53 + 5 \beta_{1} - 24 \beta_{2} ) q^{89} + ( 112 + 7 \beta_{1} + 7 \beta_{2} ) q^{91} + ( 72 + 21 \beta_{1} + 3 \beta_{2} ) q^{93} + ( -960 - 20 \beta_{1} + 60 \beta_{2} ) q^{95} + ( -36 + 24 \beta_{1} - 46 \beta_{2} ) q^{97} + ( 9 - 9 \beta_{1} - 18 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 9q^{3} + 10q^{5} + 21q^{7} + 27q^{9} + O(q^{10}) \) \( 3q + 9q^{3} + 10q^{5} + 21q^{7} + 27q^{9} + 50q^{13} + 30q^{15} + 30q^{17} + 140q^{19} + 63q^{21} - 56q^{23} + 325q^{25} + 81q^{27} - 298q^{29} + 80q^{31} + 70q^{35} - 10q^{37} + 150q^{39} + 390q^{41} + 784q^{43} + 90q^{45} - 248q^{47} + 147q^{49} + 90q^{51} - 10q^{53} - 1360q^{55} + 420q^{57} + 1500q^{59} + 810q^{61} + 189q^{63} + 860q^{65} + 1272q^{67} - 168q^{69} - 160q^{71} - 1170q^{73} + 975q^{75} - 840q^{79} + 243q^{81} + 1564q^{83} + 2740q^{85} - 894q^{87} - 178q^{89} + 350q^{91} + 240q^{93} - 2840q^{95} - 130q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 28 x + 12\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( -\nu^{2} + 7 \nu + 17 \)
\(\beta_{2}\)\(=\)\( \nu^{2} + \nu - 19 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 2\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(7 \beta_{2} - \beta_{1} + 150\)\()/8\)

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
0.424864
−5.03475
5.60988
0 3.00000 0 −15.3946 0 7.00000 0 9.00000 0
1.2 0 3.00000 0 4.31394 0 7.00000 0 9.00000 0
1.3 0 3.00000 0 21.0807 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.bv 3
4.b odd 2 1 1344.4.a.bt 3
8.b even 2 1 672.4.a.o 3
8.d odd 2 1 672.4.a.q yes 3
24.f even 2 1 2016.4.a.y 3
24.h odd 2 1 2016.4.a.z 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.4.a.o 3 8.b even 2 1
672.4.a.q yes 3 8.d odd 2 1
1344.4.a.bt 3 4.b odd 2 1
1344.4.a.bv 3 1.a even 1 1 trivial
2016.4.a.y 3 24.f even 2 1
2016.4.a.z 3 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1344))\):

\( T_{5}^{3} - 10 T_{5}^{2} - 300 T_{5} + 1400 \)
\( T_{11}^{3} - 2840 T_{11} + 45280 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( ( -3 + T )^{3} \)
$5$ \( 1400 - 300 T - 10 T^{2} + T^{3} \)
$7$ \( ( -7 + T )^{3} \)
$11$ \( 45280 - 2840 T + T^{3} \)
$13$ \( 26920 - 980 T - 50 T^{2} + T^{3} \)
$17$ \( 275880 - 6380 T - 30 T^{2} + T^{3} \)
$19$ \( -6080 - 3120 T - 140 T^{2} + T^{3} \)
$23$ \( -3285792 - 35288 T + 56 T^{2} + T^{3} \)
$29$ \( -5325704 - 6932 T + 298 T^{2} + T^{3} \)
$31$ \( 8700160 - 69600 T - 80 T^{2} + T^{3} \)
$37$ \( -4978760 - 54260 T + 10 T^{2} + T^{3} \)
$41$ \( 21358600 - 49100 T - 390 T^{2} + T^{3} \)
$43$ \( -12439552 + 183552 T - 784 T^{2} + T^{3} \)
$47$ \( 452096 + 19168 T + 248 T^{2} + T^{3} \)
$53$ \( -32747080 - 316660 T + 10 T^{2} + T^{3} \)
$59$ \( -90301120 + 659920 T - 1500 T^{2} + T^{3} \)
$61$ \( 276011400 - 372500 T - 810 T^{2} + T^{3} \)
$67$ \( -71066624 + 527328 T - 1272 T^{2} + T^{3} \)
$71$ \( 258709600 - 907800 T + 160 T^{2} + T^{3} \)
$73$ \( -337981800 - 125300 T + 1170 T^{2} + T^{3} \)
$79$ \( -29596160 + 79840 T + 840 T^{2} + T^{3} \)
$83$ \( -33022272 + 608432 T - 1564 T^{2} + T^{3} \)
$89$ \( -53515224 - 214572 T + 178 T^{2} + T^{3} \)
$97$ \( -400247720 - 1507380 T + 130 T^{2} + T^{3} \)
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